college homework math

1.1 Real Numbers: Algebra Essentials

• ⓐ 11 1 11 1
• ⓒ − 4 1 − 4 1
• ⓐ 4 (or 4.0), terminating;
• ⓑ 0. 615384 ¯ , 0. 615384 ¯ , repeating;
• ⓒ –0.85, terminating
• ⓐ rational and repeating;
• ⓑ rational and terminating;
• ⓒ irrational;
• ⓓ rational and terminating;
• ⓔ irrational
• ⓐ positive, irrational; right
• ⓑ negative, rational; left
• ⓒ positive, rational; right
• ⓓ negative, irrational; left
• ⓔ positive, rational; right

• ⓐ 11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;
• ⓑ 33, distributive property;
• ⓒ 26, distributive property;
• ⓓ 4 9 , 4 9 , commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;
• ⓔ 0, distributive property, inverse property of addition, identity property of addition

• ⓒ 121 3 π 121 3 π ;
• ⓐ −2 y −2 z or  −2 ( y + z ) ; −2 y −2 z or  −2 ( y + z ) ;
• ⓑ 2 t −1 ; 2 t −1 ;
• ⓒ 3 p q −4 p + q ; 3 p q −4 p + q ;
• ⓓ 7 r −2 s + 6 7 r −2 s + 6

A = P ( 1 + r t ) A = P ( 1 + r t )

1.2 Exponents and Scientific Notation

• ⓐ k 15 k 15
• ⓑ ( 2 y ) 5 ( 2 y ) 5
• ⓒ t 14 t 14
• ⓑ ( −3 ) 5 ( −3 ) 5
• ⓒ ( e f 2 ) 2 ( e f 2 ) 2
• ⓐ ( 3 y ) 24 ( 3 y ) 24
• ⓑ t 35 t 35
• ⓒ ( − g ) 16 ( − g ) 16
• ⓐ 1 ( −3 t ) 6 1 ( −3 t ) 6
• ⓑ 1 f 3 1 f 3
• ⓒ 2 5 k 3 2 5 k 3
• ⓐ t −5 = 1 t 5 t −5 = 1 t 5
• ⓑ 1 25 1 25
• ⓐ g 10 h 15 g 10 h 15
• ⓑ 125 t 3 125 t 3
• ⓒ −27 y 15 −27 y 15
• ⓓ 1 a 18 b 21 1 a 18 b 21
• ⓔ r 12 s 8 r 12 s 8
• ⓐ b 15 c 3 b 15 c 3
• ⓑ 625 u 32 625 u 32
• ⓒ −1 w 105 −1 w 105
• ⓓ q 24 p 32 q 24 p 32
• ⓔ 1 c 20 d 12 1 c 20 d 12
• ⓐ v 6 8 u 3 v 6 8 u 3
• ⓑ 1 x 3 1 x 3
• ⓒ e 4 f 4 e 4 f 4
• ⓓ 27 r s 27 r s
• ⓕ 16 h 10 49 16 h 10 49
• ⓐ $ 1.52 × 10 5 $ 1.52 × 10 5
• ⓑ 7.158 × 10 9 7.158 × 10 9
• ⓒ $ 8.55 × 10 13 $ 8.55 × 10 13
• ⓓ 3.34 × 10 −9 3.34 × 10 −9
• ⓔ 7.15 × 10 −8 7.15 × 10 −8
• ⓐ 703 , 000 703 , 000
• ⓑ −816 , 000 , 000 , 000 −816 , 000 , 000 , 000
• ⓒ −0.000 000 000 000 39 −0.000 000 000 000 39
• ⓓ 0.000008 0.000008
• ⓐ − 8.475 × 10 6 − 8.475 × 10 6
• ⓑ 8 × 10 − 8 8 × 10 − 8
• ⓒ 2.976 × 10 13 2.976 × 10 13
• ⓓ − 4.3 × 10 6 − 4.3 × 10 6
• ⓔ ≈ 1.24 × 10 15 ≈ 1.24 × 10 15

Number of cells: 3 × 10 13 ; 3 × 10 13 ; length of a cell: 8 × 10 −6 8 × 10 −6 m; total length: 2.4 × 10 8 2.4 × 10 8 m or 240 , 000 , 000 240 , 000 , 000 m.

1.3 Radicals and Rational Exponents

5 | x | | y | 2 y z . 5 | x | | y | 2 y z . Notice the absolute value signs around x and y ? That’s because their value must be positive!

10 | x | 10 | x |

x 2 3 y 2 . x 2 3 y 2 . We do not need the absolute value signs for y 2 y 2 because that term will always be nonnegative.

b 4 3 a b b 4 3 a b

14 −7 3 14 −7 3

• ⓒ 88 9 3 88 9 3

( 9 ) 5 = 3 5 = 243 ( 9 ) 5 = 3 5 = 243

x ( 5 y ) 9 2 x ( 5 y ) 9 2

28 x 23 15 28 x 23 15

1.4 Polynomials

The degree is 6, the leading term is − x 6 , − x 6 , and the leading coefficient is −1. −1.

2 x 3 + 7 x 2 −4 x −3 2 x 3 + 7 x 2 −4 x −3

−11 x 3 − x 2 + 7 x −9 −11 x 3 − x 2 + 7 x −9

3 x 4 −10 x 3 −8 x 2 + 21 x + 14 3 x 4 −10 x 3 −8 x 2 + 21 x + 14

3 x 2 + 16 x −35 3 x 2 + 16 x −35

16 x 2 −8 x + 1 16 x 2 −8 x + 1

4 x 2 −49 4 x 2 −49

6 x 2 + 21 x y −29 x −7 y + 9 6 x 2 + 21 x y −29 x −7 y + 9

1.5 Factoring Polynomials

( b 2 − a ) ( x + 6 ) ( b 2 − a ) ( x + 6 )

( x −6 ) ( x −1 ) ( x −6 ) ( x −1 )

• ⓐ ( 2 x + 3 ) ( x + 3 ) ( 2 x + 3 ) ( x + 3 )
• ⓑ ( 3 x −1 ) ( 2 x + 1 ) ( 3 x −1 ) ( 2 x + 1 )

( 7 x −1 ) 2 ( 7 x −1 ) 2

( 9 y + 10 ) ( 9 y − 10 ) ( 9 y + 10 ) ( 9 y − 10 )

( 6 a + b ) ( 36 a 2 −6 a b + b 2 ) ( 6 a + b ) ( 36 a 2 −6 a b + b 2 )

( 10 x − 1 ) ( 100 x 2 + 10 x + 1 ) ( 10 x − 1 ) ( 100 x 2 + 10 x + 1 )

( 5 a −1 ) − 1 4 ( 17 a −2 ) ( 5 a −1 ) − 1 4 ( 17 a −2 )

1.6 Rational Expressions

1 x + 6 1 x + 6

( x + 5 ) ( x + 6 ) ( x + 2 ) ( x + 4 ) ( x + 5 ) ( x + 6 ) ( x + 2 ) ( x + 4 )

2 ( x −7 ) ( x + 5 ) ( x −3 ) 2 ( x −7 ) ( x + 5 ) ( x −3 )

x 2 − y 2 x y 2 x 2 − y 2 x y 2

1.1 Section Exercises

irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.

The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

−14 y − 11 −14 y − 11

−4 b + 1 −4 b + 1

43 z − 3 43 z − 3

9 y + 45 9 y + 45

−6 b + 6 −6 b + 6

16 x 3 16 x 3

1 2 ( 40 − 10 ) + 5 1 2 ( 40 − 10 ) + 5

irrational number

g + 400 − 2 ( 600 ) = 1200 g + 400 − 2 ( 600 ) = 1200

inverse property of addition

1.2 Section Exercises

No, the two expressions are not the same. An exponent tells how many times you multiply the base. So 2 3 2 3 is the same as 2 × 2 × 2 , 2 × 2 × 2 , which is 8. 3 2 3 2 is the same as 3 × 3 , 3 × 3 , which is 9.

It is a method of writing very small and very large numbers.

12 40 12 40

1 7 9 1 7 9

3.14 × 10 − 5 3.14 × 10 − 5

16,000,000,000

b 6 c 8 b 6 c 8

a b 2 d 3 a b 2 d 3

q 5 p 6 q 5 p 6

y 21 x 14 y 21 x 14

72 a 2 72 a 2

c 3 b 9 c 3 b 9

y 81 z 6 y 81 z 6

1.0995 × 10 12 1.0995 × 10 12

0.00000000003397 in.

12,230,590,464 m 66 m 66

a 14 1296 a 14 1296

n a 9 c n a 9 c

1 a 6 b 6 c 6 1 a 6 b 6 c 6

0.000000000000000000000000000000000662606957

1.3 Section Exercises

When there is no index, it is assumed to be 2 or the square root. The expression would only be equal to the radicand if the index were 1.

The principal square root is the nonnegative root of the number.

9 5 5 9 5 5

6 10 19 6 10 19

− 1 + 17 2 − 1 + 17 2

7 2 3 7 2 3

20 x 2 20 x 2

17 m 2 m 17 m 2 m

2 b a 2 b a

15 x 7 15 x 7

5 y 4 2 5 y 4 2

4 7 d 7 d 4 7 d 7 d

2 2 + 2 6 x 1 −3 x 2 2 + 2 6 x 1 −3 x

− w 2 w − w 2 w

3 x − 3 x 2 3 x − 3 x 2

5 n 5 5 5 n 5 5

9 m 19 m 9 m 19 m

2 3 d 2 3 d

3 2 x 2 4 2 3 2 x 2 4 2

6 z 2 3 6 z 2 3

−5 2 −6 7 −5 2 −6 7

m n c a 9 c m n m n c a 9 c m n

2 2 x + 2 4 2 2 x + 2 4

1.4 Section Exercises

The statement is true. In standard form, the polynomial with the highest value exponent is placed first and is the leading term. The degree of a polynomial is the value of the highest exponent, which in standard form is also the exponent of the leading term.

Use the distributive property, multiply, combine like terms, and simplify.

4 x 2 + 3 x + 19 4 x 2 + 3 x + 19

3 w 2 + 30 w + 21 3 w 2 + 30 w + 21

11 b 4 −9 b 3 + 12 b 2 −7 b + 8 11 b 4 −9 b 3 + 12 b 2 −7 b + 8

24 x 2 −4 x −8 24 x 2 −4 x −8

24 b 4 −48 b 2 + 24 24 b 4 −48 b 2 + 24

99 v 2 −202 v + 99 99 v 2 −202 v + 99

8 n 3 −4 n 2 + 72 n −36 8 n 3 −4 n 2 + 72 n −36

9 y 2 −42 y + 49 9 y 2 −42 y + 49

16 p 2 + 72 p + 81 16 p 2 + 72 p + 81

9 y 2 −36 y + 36 9 y 2 −36 y + 36

16 c 2 −1 16 c 2 −1

225 n 2 −36 225 n 2 −36

−16 m 2 + 16 −16 m 2 + 16

121 q 2 −100 121 q 2 −100

16 t 4 + 4 t 3 −32 t 2 − t + 7 16 t 4 + 4 t 3 −32 t 2 − t + 7

y 3 −6 y 2 − y + 18 y 3 −6 y 2 − y + 18

3 p 3 − p 2 −12 p + 10 3 p 3 − p 2 −12 p + 10

a 2 − b 2 a 2 − b 2

16 t 2 −40 t u + 25 u 2 16 t 2 −40 t u + 25 u 2

4 t 2 + x 2 + 4 t −5 t x − x 4 t 2 + x 2 + 4 t −5 t x − x

24 r 2 + 22 r d −7 d 2 24 r 2 + 22 r d −7 d 2

32 x 2 −4 x −3 32 x 2 −4 x −3 m 2

32 t 3 − 100 t 2 + 40 t + 38 32 t 3 − 100 t 2 + 40 t + 38

a 4 + 4 a 3 c −16 a c 3 −16 c 4 a 4 + 4 a 3 c −16 a c 3 −16 c 4

1.5 Section Exercises

The terms of a polynomial do not have to have a common factor for the entire polynomial to be factorable. For example, 4 x 2 4 x 2 and −9 y 2 −9 y 2 don’t have a common factor, but the whole polynomial is still factorable: 4 x 2 −9 y 2 = ( 2 x + 3 y ) ( 2 x −3 y ) . 4 x 2 −9 y 2 = ( 2 x + 3 y ) ( 2 x −3 y ) .

Divide the x x term into the sum of two terms, factor each portion of the expression separately, and then factor out the GCF of the entire expression.

10 m 3 10 m 3

( 2 a −3 ) ( a + 6 ) ( 2 a −3 ) ( a + 6 )

( 3 n −11 ) ( 2 n + 1 ) ( 3 n −11 ) ( 2 n + 1 )

( p + 1 ) ( 2 p −7 ) ( p + 1 ) ( 2 p −7 )

( 5 h + 3 ) ( 2 h −3 ) ( 5 h + 3 ) ( 2 h −3 )

( 9 d −1 ) ( d −8 ) ( 9 d −1 ) ( d −8 )

( 12 t + 13 ) ( t −1 ) ( 12 t + 13 ) ( t −1 )

( 4 x + 10 ) ( 4 x − 10 ) ( 4 x + 10 ) ( 4 x − 10 )

( 11 p + 13 ) ( 11 p − 13 ) ( 11 p + 13 ) ( 11 p − 13 )

( 19 d + 9 ) ( 19 d − 9 ) ( 19 d + 9 ) ( 19 d − 9 )

( 12 b + 5 c ) ( 12 b − 5 c ) ( 12 b + 5 c ) ( 12 b − 5 c )

( 7 n + 12 ) 2 ( 7 n + 12 ) 2

( 15 y + 4 ) 2 ( 15 y + 4 ) 2

( 5 p − 12 ) 2 ( 5 p − 12 ) 2

( x + 6 ) ( x 2 − 6 x + 36 ) ( x + 6 ) ( x 2 − 6 x + 36 )

( 5 a + 7 ) ( 25 a 2 − 35 a + 49 ) ( 5 a + 7 ) ( 25 a 2 − 35 a + 49 )

( 4 x − 5 ) ( 16 x 2 + 20 x + 25 ) ( 4 x − 5 ) ( 16 x 2 + 20 x + 25 )

( 5 r + 12 s ) ( 25 r 2 − 60 r s + 144 s 2 ) ( 5 r + 12 s ) ( 25 r 2 − 60 r s + 144 s 2 )

( 2 c + 3 ) − 1 4 ( −7 c − 15 ) ( 2 c + 3 ) − 1 4 ( −7 c − 15 )

( x + 2 ) − 2 5 ( 19 x + 10 ) ( x + 2 ) − 2 5 ( 19 x + 10 )

( 2 z − 9 ) − 3 2 ( 27 z − 99 ) ( 2 z − 9 ) − 3 2 ( 27 z − 99 )

( 14 x −3 ) ( 7 x + 9 ) ( 14 x −3 ) ( 7 x + 9 )

( 3 x + 5 ) ( 3 x −5 ) ( 3 x + 5 ) ( 3 x −5 )

( 2 x + 5 ) 2 ( 2 x − 5 ) 2 ( 2 x + 5 ) 2 ( 2 x − 5 ) 2

( 4 z 2 + 49 a 2 ) ( 2 z + 7 a ) ( 2 z − 7 a ) ( 4 z 2 + 49 a 2 ) ( 2 z + 7 a ) ( 2 z − 7 a )

1 ( 4 x + 9 ) ( 4 x −9 ) ( 2 x + 3 ) 1 ( 4 x + 9 ) ( 4 x −9 ) ( 2 x + 3 )

1.6 Section Exercises

You can factor the numerator and denominator to see if any of the terms can cancel one another out.

True. Multiplication and division do not require finding the LCD because the denominators can be combined through those operations, whereas addition and subtraction require like terms.

y + 5 y + 6 y + 5 y + 6

3 b + 3 3 b + 3

x + 4 2 x + 2 x + 4 2 x + 2

a + 3 a − 3 a + 3 a − 3

3 n − 8 7 n − 3 3 n − 8 7 n − 3

c − 6 c + 6 c − 6 c + 6

d 2 − 25 25 d 2 − 1 d 2 − 25 25 d 2 − 1

t + 5 t + 3 t + 5 t + 3

6 x − 5 6 x + 5 6 x − 5 6 x + 5

p + 6 4 p + 3 p + 6 4 p + 3

2 d + 9 d + 11 2 d + 9 d + 11

12 b + 5 3 b −1 12 b + 5 3 b −1

4 y −1 y + 4 4 y −1 y + 4

10 x + 4 y x y 10 x + 4 y x y

9 a − 7 a 2 − 2 a − 3 9 a − 7 a 2 − 2 a − 3

2 y 2 − y + 9 y 2 − y − 2 2 y 2 − y + 9 y 2 − y − 2

5 z 2 + z + 5 z 2 − z − 2 5 z 2 + z + 5 z 2 − z − 2

x + 2 x y + y x + x y + y + 1 x + 2 x y + y x + x y + y + 1

2 b + 7 a a b 2 2 b + 7 a a b 2

18 + a b 4 b 18 + a b 4 b

a − b a − b

3 c 2 + 3 c − 2 2 c 2 + 5 c + 2 3 c 2 + 3 c − 2 2 c 2 + 5 c + 2

15 x + 7 x −1 15 x + 7 x −1

x + 9 x −9 x + 9 x −9

1 y + 2 1 y + 2

Review Exercises

y = 24 y = 24

3 a 6 3 a 6

x 3 32 y 3 x 3 32 y 3

1.634 × 10 7 1.634 × 10 7

4 2 5 4 2 5

7 2 50 7 2 50

3 x 3 + 4 x 2 + 6 3 x 3 + 4 x 2 + 6

5 x 2 − x + 3 5 x 2 − x + 3

k 2 − 3 k − 18 k 2 − 3 k − 18

x 3 + x 2 + x + 1 x 3 + x 2 + x + 1

3 a 2 + 5 a b − 2 b 2 3 a 2 + 5 a b − 2 b 2

4 a 2 4 a 2

( 4 a − 3 ) ( 2 a + 9 ) ( 4 a − 3 ) ( 2 a + 9 )

( x + 5 ) 2 ( x + 5 ) 2

( 2 h − 3 k ) 2 ( 2 h − 3 k ) 2

( p + 6 ) ( p 2 − 6 p + 36 ) ( p + 6 ) ( p 2 − 6 p + 36 )

( 4 q − 3 p ) ( 16 q 2 + 12 p q + 9 p 2 ) ( 4 q − 3 p ) ( 16 q 2 + 12 p q + 9 p 2 )

( p + 3 ) 1 3 ( −5 p − 24 ) ( p + 3 ) 1 3 ( −5 p − 24 )

x + 3 x − 4 x + 3 x − 4

m + 2 m − 3 m + 2 m − 3

6 x + 10 y x y 6 x + 10 y x y

Practice Test

x = –2 x = –2

3 x 4 3 x 4

13 q 3 − 4 q 2 − 5 q 13 q 3 − 4 q 2 − 5 q

n 3 − 6 n 2 + 12 n − 8 n 3 − 6 n 2 + 12 n − 8

( 4 x + 9 ) ( 4 x − 9 ) ( 4 x + 9 ) ( 4 x − 9 )

( 3 c − 11 ) ( 9 c 2 + 33 c + 121 ) ( 3 c − 11 ) ( 9 c 2 + 33 c + 121 )

4 z − 3 2 z − 1 4 z − 3 2 z − 1

3 a + 2 b 3 b 3 a + 2 b 3 b

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Choose Your Test

Sat / act prep online guides and tips, the 5 best homework help websites (free and paid).

Other High School , General Education

Listen: we know homework isn’t fun, but it is a good way to reinforce the ideas and concepts you’ve learned in class. But what if you’re really struggling with your homework assignments?

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• The pros and cons of using these websites for homework help 
• The line between “learning” and “cheating” when using online homework help 
• Tips for getting the most out of a homework help website

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The Basics About Homework Help Websites–Free and Paid

Homework help websites are designed to help you complete your homework assignments, plain and simple. 

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You don't need a golden piggy bank to use homework help websites. Some provide low or no cost help for students like you!

How Expensive Are the Best Homework Help Websites?

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The 5 Best Homework Help Websites 

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The best homework help websites are the ones that are reliable and help you learn the material. They don’t just provide answers to homework questions–they actually help you learn the material. 

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Best Free Homework Help Site: Khan Academy

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Runner Up: Brainly.com offers a free service that allows you to type in questions and get answers and explanations from experts. The downside is that you’re limited to two answers per question and have to watch ads. 

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Runner Up: KhanAcademy offers in-depth tutorials that cover complex math topics for free, but you won’t get the same tailored help (and answers!) that Photomath offers. 

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We’ll go over the good–and the not-so-good–aspects of getting online homework help below. 

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Not everyone learns the same way. Maybe you’re more of a visual learner, but your teacher mostly does lectures. Or maybe you learn best by listening and taking notes, but you’re expected to learn something just from reading the textbook . 

One of the best things about online homework help is that it comes in a variety of forms. The best homework help sites offer resources for all types of learners, including videos, practice activities, and even one-on-one discussions with real-life experts. 

This variety can also be a good thing if you just don’t really resonate with the way a concept is being explained (looking at you, math textbooks!).

Not so fast. There are cons to homework help websites, too. Get to know them below!

3 Cons of Using Homework Help Websites 

Now, let’s take a look at the drawbacks of online homework help. 

#1: Unreliable Info

This can be a real problem. In addition to all the really good homework help sites, there are a whole lot of disreputable or unreliable sites out there. The fact of the matter is that some homework help sites don’t necessarily hire people who are experts in the subjects they’re talking about. In those cases, you may not be getting the accurate, up-to-date, and thorough information you need.

Additionally, even the great sites may not be able to answer all of your homework questions. This is especially true if the site uses an algorithm or chatbot to help students…or if you’re enrolled in an advanced or college-level course. In these cases, working with your teacher or school-provided tutors are probably your best option. 

#2: No Clarification

This depends on the service you use, of course. But the majority of them provide free or low-cost help through pre-recorded videos. Watching videos or reading info online can definitely help you with your homework… but you can’t ask questions or get immediate feedback if you need it .

#3: Potential For Scamming 

Like we mentioned earlier, there are a lot of homework help websites out there, and lots of them are scams. The review comments we read covered everything from outdated or wrong information, to misleading claims about the help provided, to not allowing people to cancel their service after signing up. 

No matter which site you choose to use, make sure you research and read reviews before you sign up–especially if it’s a paid service! 

When Does “Help” Become “Cheating”?

Admittedly, whether using homework help websites constitutes cheating is a bit of a grey area. For instance, is it “help” when a friend reads your essay for history class and corrects your grammar, or is it “cheating”? The truth is, not everyone agrees on when “help” crosses the line into “cheating .” When in doubt, it can be a good idea to check with your teacher to see what they think about a particular type of help you want to get. 

That said, a general rule of thumb to keep in mind is to make sure that the assignment you turn in for credit is authentically yours . It needs to demonstrate your own thoughts and your own current abilities. Remember: the point of every homework assignment is to 1) help you learn something, and 2) show what you’ve learned. 

So if a service answers questions or writes essays for you, there’s a good chance using it constitutes cheating. 

Here’s an example that might help clarify the difference for you. Brainstorming essay ideas with others or looking online for inspiration is “help” as long as you write the essay yourself. Having someone read it and give you feedback about what you need to change is also help, provided you’re the one that makes the changes later. 

But copying all or part of an essay you find online or having someone write (or rewrite) the whole thing for you would be “cheating.” The same is true for other subjects. Ultimately, if you’re not generating your own work or your own answers, it’s probably cheating.

5 Tips for Finding the Best Homework Help Websites for You

Now that you know some of our favorite homework help websites, free and paid, you can start doing some additional research on your own to decide which services might work best for you! Here are some top tips for choosing a homework help website. 

Tip 1: Decide How You Learn Best 

Before you decide which site or sites you’re going to use for homework help, y ou should figure out what kind of learning style works for you the most. Are you a visual learner? Then choose a site that uses lots of videos to help explain concepts. If you know you learn best by actually doing tasks, choose a site that provides lots of practice exercises.

Tip 2: Determine Which Subjects You Need Help With

Just because a homework help site is good overall doesn’t mean that it’s equally good for every subject. If you only need help in math, choose a site that specializes in that area. But if history is where you’re struggling, a site that specializes in math won’t be much help. So make sure to choose a site that you know provides high-quality help in the areas you need it most. 

Tip 3: Decide How Much One-On-One Help You Need 

This is really about cost-effectiveness. If you learn well on your own by reading and watching videos, a free site like Khan Academy is a good choice. But if you need actual tutoring, or to be able to ask questions and get personalized answers from experts, a paid site that provides that kind of service may be a better option.

Tip 4: Set a Budget

If you decide you want to go with a paid homework help website, set a budget first . The prices for sites vary wildly, and the cost to use them can add up quick. 

Tip 5: Read the Reviews

Finally, it’s always a good idea to read actual reviews written by the people using these homework sites. You’ll learn the good, the bad, and the ugly of what the users’ experiences have been. This is especially true if you intend to subscribe to a paid service. You’ll want to make sure that users think it’s worth the price overall!

What’s Next?

If you want to get good grades on your homework, it’s a good idea to learn how to tackle it strategically. Our expert tips will help you get the most out of each assignment…and boost your grades in the process.

Doing well on homework assignments is just one part of getting good grades. We’ll teach you everything you need to know about getting great grades in high school in this article.

Of course, test grades can make or break your GPA, too. Here are 17 expert tips that’ll help you get the most out of your study prep before you take an exam.

Ashley Sufflé Robinson has a Ph.D. in 19th Century English Literature. As a content writer for PrepScholar, Ashley is passionate about giving college-bound students the in-depth information they need to get into the school of their dreams.

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College Algebra

Also known as "High School Algebra"

OK. So what are you going to learn here?

You will learn about Numbers, Polynomials, Inequalities, Sequences and Sums, many types of Functions, and how to solve them.

You will also gain a deeper insight into Mathematics, get to practice using your new skills with lots of examples and questions, and generally improve your mind.

With your new skills you will be able to put together mathematical models so you can find good quality solutions to many tricky real world situations.

Near the end of most pages is a "Your Turn" section ... do these! You need to balance your reading with doing . Answering questions helps you sort things out in your mind. And don't guess the answer: use pen and paper and try your best before seeing the solution.

So what is this thing called Mathematics? And how do you go about learning it?

• Welcome to Mathematics
• Learning Mathematics
• The Language of Mathematics
• Symbols in Algebra

Next, we need to think about mathematics in terms of sets .

• Introduction to Sets

Now we know what a set is, let us look at different sets of numbers that are useful:

• The Evolution of Numbers
• Prime and Composite Numbers
• Fundamental Theorem of Arithmetic
• Whole Numbers and Integers
• Rational Numbers
• Using Rational Numbers
• Irrational Numbers
• 0.999... = 1
• Real Numbers
• Imaginary Numbers
• Complex Numbers
• Multiplying Complex Numbers
• The Complex Plane
• Common Number Sets

Inequalities

"Equal To" is nice but not always available. Maybe we only know that something is less than, or greater than. So let's learn about in equalities.

• Introduction to Inequalities
• Properties of Inequalities
• Solving Inequalities
• Solving Inequality Word Questions

We will be using exponents a lot, so let's get to know them well.

• Variables with Exponents
• Using Exponents in Algebra
• Squares and Square Roots
• Squares and Square Roots in Algebra
• Fractional Exponents
• Laws of Exponents
• Exponents of Negative Numbers

Polynomials

Polynomials were some of the first things ever studied in Algebra. They are simple, yet powerful in their ability to model real world situations.

• What is a Polynomial?
• Adding And Subtracting Polynomials
• Multiplying Polynomials
• Polynomials - Long Multiplication
• Dividing Polynomials
• Polynomials - Long Division
• Degree (of an Expression)
• Special Binomial Products
• Difference of Two Cubes
• Factoring in Algebra
• Solving Polynomials
• Roots of Polynomials: Sums and Products
• Rational Expressions
• Using Rational Expressions
• Fundamental Theorem of Algebra
• Remainder Theorem and Factor Theorem
• General Form of a Polynomial

Graphing Polynomials

• How Polynomials Behave
• Polynomials: The Rule of Signs
• Polynomials: Bounds on Zeros

And, of course, we need to know about equations ... and how to solve them.

• Equations and Formulas
• Solving Equations
• Solving Word Questions
• Zero Product Property
• Implication and Iff
• Theorems, Corollaries, Lemmas

Graphs can save us! They are a great way to see what is going on and can help us solve many things. But we need to be careful, as they sometimes don't give the full story.

• Cartesian Coordinates
• Pythagoras' Theorem
• Distance Between 2 Points
• Graph of an Equation
• Finding Intercepts From an Equation
• Symmetry in Equations
• Linear Equations

They are just equations for lines. But they come in many forms.

• Equation of a Straight Line
• Point-Slope Equation of a Line
• General Form of Equation of a Line
• Equation of a Line from 2 Points
• Midpoint of a Line Segment
• Parallel and Perpendicular Lines

A function relates an input to an output. But from that simple foundation many useful things can be built.

• What is a Function?
• Domain, Range and Codomain
• Evaluating Functions
• Increasing and Decreasing Functions
• Maxima and Minima of Functions
• Even and Odd Functions
• Set-Builder Notation

Common Functions Reference :

• Square Function
• Square Root Function
• Cube Function
• Reciprocal Function
• Absolute Value Function
• Floor and Ceiling Function
• Function Transformations
• Equation Grapher
• Operations with Functions
• Composition of Functions
• Inverse Functions

Equations of Second Degree

"Second degree" just means the variable has an exponent of 2, like x 2 . It is the next major step after linear equations (where the exponent is 1, like x).

• Quadratic Equations
• Factoring Quadratics
• Completing the Square
• Derivation of Quadratic Formula
• Graphing Quadratic Equations
• Quadratic Equations in the Real World
• Circle Equations

We already have experience in solving, but now we can learn more!

• Mathematical Models and Mathematical Models 2
• Approximate Solutions
• Intermediate Value Theorem
• Solving Radical Equations
• Change of Variables
• Algebra Mistakes

We learned about inequalities above, now let's learn how to solve them.

• Graphing Linear Inequalities
• Inequality Graphing Tool
• Solving Quadratic Inequalities
• Solving Rational Inequalities
• Absolute Value in Algebra

Exponents and Logarithms

We already know about exponents ... well logarithms just go the other way. And together they can be very powerful.

• Introduction to Logarithms
• Exponents, Roots and Logarithms
• Working with Exponents and Logarithms
• Exponential Function
• Logarithmic Function
• Exponential Growth and Decay
• Systems of Linear Equations

What happens when we have two or more linear equations that work together? They can often be solved! It isn't very hard but can take a lot of calculations.

• Types of Matrix
• How to Multiply Matrices
• Determinant of a Matrix
• Inverse of a Matrix:
• Using Elementary Row Operations (Gauss-Jordan)
• Using Minors, Cofactors and Adjugate
• Scalar, Vector, Matrix and Vectors
• Matrix Calculator
• More at Matrix Index
• Solving Systems of Linear Equations Using Matrices
• Systems of Linear and Quadratic Equations
• Probability

Is it likely? You be the judge!

• The Basic Counting Principle
• Combinations and Permutations

Sequences, Series and Partial Sums

A Sequence is a set of things (usually numbers) that are in order. We can also sum up a series, where Sigma Notation is very useful.

• Sequences - Finding A Rule
• Sigma Notation
• Partial Sums
• Arithmetic Sequences and Sums
• Geometric Sequences and Sums

These last few subjects use what we have learned above.

• Partial Fractions
• Mathematical Induction
• Pascal's Triangle
• Binomial Theorem

And that is all!

But there are many other interesting algebra topics such as:

• Euler's Formula for Complex Numbers
• Taylor Series (needs a basic understanding of derivatives )

Math Help Resources

Mathcracker.com.

Our site offers a wide variety of Math Help resources , including hundreds of Statistics, Calculus, Algebra, Decision Analysis and Finance calculators with steps. So please search around to find what you need. We are continuously adding new tutorials and lessons, solvers, online calculators and solved math problems.

The math help we provide is mostly suitable for college and high school students , even though we believe that there is a little bit for everyone. We can help you with the following subjects:

Search our Math and Stats Help tutorials, online solvers and calculators with the navigation menus on the top of the page, or you can search what you need in the search box below:

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Mathcracker aims to help YOU LEARN math by giving you access to learning tools. Our belief is that Math is fundamental to understanding and functioning in the modern world, plus, math is also pretty neat. Our desire is to make math accessible to everyone in an open-access, democratic environment, by helping you to learn – and do – math using our tools, solvers, articles.

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Finally! We are very pleased to announce the new version of the site. We have added new exciting features such a crisp new graph designs, with a thorough review of the content and materials.

● And we have added a much anticipated feature: now you will be able to download the step-by-solutions our calculators provide in a Word (docx) file.

● We have also added a premium membership that allows you to get access to our content and solver/calculators, without any ads.

● Also, our members will have access to a 30,000+ collection of step-by-step sample solved Math and Stats problems.

● And many other features to be unveiled throughout the year, including artificial intelligence, automated worksheets, lessons and practice, along with automated detection and solution of problems.

This year is the year where MathCracker will make a substantial leap in terms of the content offered in the site. So, keep tuned for more math solvers, calculators and lessons, and all the success in this new academic year.

This in depth and meaty tutorial will help you understand how to factor algebraic expressions .

Simplifying radicals gotta be THE most useful thing to know about in Basic Algebra. Learn about it in this tutorial .

Among all things related to the rules of exponents, students tend to get confused when dealing with negative exponents . This tutorial will clarify all your doubts.

One of the crucial skills required to deal with algebraic expression is the proper use of rules of exponents .

All you need to know about Cramer's Rule and how to use it to solve systems of linear equations.

See this tutorial to learn how to and when to use the mean, median or mode as the measure of center, depending on the type of distribution.

What is a Box and Whisker Plot? How do you define the box? The Whiskers? Find out in this tutorial .

And to conclude the series on conic sections, it is now time for the ellipse .

We continue our series of tutorial on conic sections. This time, it is the turn of the hyperbola .

Conic sections and the general equation of the parabola .

A very complete and in depth tutorial on partial fractions decompositions , with concrete steps to follow.

A complete tutorial of how to apply log rules , with explanations and examples.

Take a look at this tutorial about PEMDAS and the convention for the order of operations .

Learn more about all the concepts related with the graph of a function .

One of the most versatile Math artifacts: See how the unit circle crosses Algebra, Geometry and Trigonometry.

Crucial concepts when dealing with functions: Domain and Range .

Another great, in-depth tutorial. All you need to know about the Absolute Value of a number .

Here is another crucial property in Algebra: the Commutative Property .

The associative property is one of those properties that is used all the time, but we do not really know.

All there is to know about horizontal asymptotes .

Learn all that there is to the circle: The Equation of a Circle .

We have started a new section for Finance solvers. Check them out: Future Value Calculator , Present Value Calculator , Present Value of an Annuity , among many other Finance Calculators .

One of the best advice right before starting a new semester: Don't panic .

Excellent article about why it is usually the easy Stats classes those where students need the most help with. Check it out! .

Going for a large overhaul in the site, which much more powerful tools. In the meantime, here I leave you a new article Tricks of the Trade: Finding Stats Homework Help . Another interesting post about why more and more students end up learning their Stats using private tutors. Check it here . Also all, you need to know about the process finding help with your finance assignment .

Although we already had a complete descriptive statistics calculator , I have requests to make some specific calculators, to compute one measure at a time, so here I have it: I added a mean calculator , standard deviation calculator and median calculator .

I have added a page that puts together all the articles we have available about education topics . Also, I found this interesting article about the advantages of staring with Algebra early in life.

Check our revised SPSS tutorial .

I have added a couple of algebraic solvers for arithmetic sequences and geometric sequences .

Z-scores are the way we compute normal probabilities, as any general normal distribution can always be reduced to a standard normal distribution. Check out this Z-Scores Calculator .

I have found two excellent tutorials one about hypothesis testing , and the other one about statistics and parameters . Both are great reads.

I have added several effect size calculators, including a Gamma Calculator and a Phi Coefficient Calculator .

The idea of "effect size" is very relevant in statistics. If often happens that a test statistics is significant, but the association has no practical meaning. This is because a extremely large sample size alone can find significant n extremely small effect. Cramer's V is a statistic that assess the effect size or practical significance of the association between nominal variables. Check it out: Cramer's V Calculator .

More solvers for our series of elementary geometry solvers, and we now add a calculator of the area and volume of a square pyramid that will show you all the steps and formulas used to calculate the surface area and volume of a square pyramid of side \(l\) and height \(h\).

Keeping the momentum up, here you have a calculator of the area and volume of a cone that will show you all the steps and formulas used to calculate the area and volume of a right circle cone radius \(r\) and height \(h\).

Another solver for our Geometry series, a calculator of the area and volume of a sphere that will show you all the steps and formulas used to calculate the area and volume of a sphere of radius \(r\).

Typically found in special tables, here you have a critical correlation calculator . The advantage of this calculator is that it will compute the critical correlation for any value of \(n\), which is not always possible when you use critical correlation tables, which only have a few values of \(n\).

Lots of people were asking for geometry solvers and here they are. I have started a new section for Geometry and I'll be adding more solver on a continuous basis. For now, here I leave for you a area and perimeter of a circle calculator , and an area and volume calculator for cylinders

This one is a solver that user were clamoring for: Let me introduce our Polynomial Operations Calculator , with which you can add, subtract and multiply polynomials, with the solution shown step-by-step. Also, we have started with MathQuill, and excellent option for enhancing the input of math.

We expand even more our Algebra series with some more basic solvers, in this case, a absolute value calculator .

New solver in our Algebra series, now I have added a solver that evaluates all sort of algebraic expressions that can include any kind of operation (sum, subtraction, multiplication, division), numbers, fractions, trigonometric functions, exponential functions, powers, roots, etc. Check our Algebraic Expression Calculator .

Continuing the series of Algebra solvers, I have added a solver that calculates operations between fractions and shows all the steps. Depending of the operation required it conducts the appropriate steps to reach the final fraction result, in reduced form. Check out our fraction operation solver .

I have made a new function grapher to replace the old flash grapher we had. This time, I used the Google Chart format, that looks more modern and goes along the trend of JS charts.

I get a lots of requests for this one, so here it goes. This is a fraction reduction tool fraction-reduction-lowest-terms.php , that takes a fraction into its lowest terms.

This time, I added a greatest common divisor , or as it usually known as greatest common factor calculator . I took care to provide as much detail as possible in the calculations shown by the solver.

Today I start with a new series of Algebra solvers. I have added a prime decomposition calculator , you type in a positive integer number and the solver tell you in a detailed way what is its prime decomposition.

I am very pleased to announce that we have rolled a new site design, with a much improved mobile functionality.

This time I decided to write on functions and how to deal with them .

Another article. This time dealing with the possibility of having a software that can solve Math and Stats problems FULLY automatically Check Here .

I loved the feedback I got from the first Math Crack, so I decided to pump up the series. I'll be uploading them periodically. This next first Math Crack is about an easy explanation of derivatives .

I finally started a project that I have been wanting to undertake for a while. It is the Math Cracks, which is the way I call to this short tutorials that attempt to explain hard math concepts in simple terms, step-by-step. The first Math Crack of the series is Integration by Parts .

A site update is coming soon. I will be changing the layout a bit so that to keep the solvers in a more organized way and the articles will be organized rather in blog format. I will be adding a complete set of Algebra solvers, so stay tuned.

A couple of tutorials and solvers were added: Quadratic Formula: The meaning of the term -b/2a , an article about Algebra 2 and a Line Plot Maker

I have added a couple of new articles for your reading enjoyment: Math Articles

I have added a minimum sample size calculator for mean and proportion

After long working hours, I've finished all the solvers required to cover all basic statistics topics. The new solvers are: Z-test for two means , Z-test for two proportions , T-test for two means , T-test for paired samples , Chi-Square Goodness of fit , and Chi-Square test of Independence . I also added the most important non-parametric tests, which are the Sign Test , also Wilcoxon Signed Ranks Test , as well as the Wilcoxon Rank-Sum Test and Kruskal-Wallis

I get asked to review different online curricula. Thanks to a guest writer, I just added a great article about online accounting curricula. You can check it here .

Found a very informational article about the do's and dont's of Accounting Homework Help. You can check it here .

Great changes to improve user experience are coming very soon. I have written recently many math and stats solvers that I'll be adding to the site. I've been polishing the scripts we already have and this is almost turning into a full time job. As many users are aware, we have had ads for the last few months, and even though I don't like ads, it is the only way to support the site, which is turning into a full time job by now. I'm looking for a more efficient way of using ads so that the site can be ad-supported and it continue being free that way.

More Stats tests done. I have finished a t-test for one population mean with unknown population standard deviation and a Chi-Square test for one population variance . Pretty much as the other solvers I have added, they are simple to use: define hypotheses, and parameters of the test, and a detailed step-by-step calculation is performed.

I've been working on these for a while and I finally finished them. I added a z-test for one population mean and a z-test for one population proportion . Very user friendly, define hypotheses, and parameters of the test, and a detailed step-by-step calculation is performed. Recommended.

I just added a working histogram maker . You enter the dataset and the number of bins and a properly scaled histogram is rendered by the script.

I have added a very cool bar graph maker . You enter the categories, the values associated with each of the categories, and a bar chart rendered by the script.

Not exactly the typical calculator we do, but I have had some requests for it, so I added a simple mortgage calculator tool to compute the monthly payment for a mortgage, under basic assumptions (without considering PMI and insurance).

I have uploaded a 2x2 system of linear equations solver.

I have updated the pie chart maker script.

I have added two confidence interval calculators. They are confidence intervals for the population mean μ when the population standard deviation σ is known and for when the and a population standard deviation is not known .

We have two more probability calculators. They are a binomial distribution calculator and a Poisson distribution calculator .

I have added two new probability calculators, based on a couple of requests I had. They are a exponential distribution calculator and a uniform distribution calculator .

I have added a new article. Check it out here .

I have added a correlation coefficient calculator . This time, the script is more user friendly, and checks for errors in the data before processing them.

I received a whole lot of e-mails asking for a tool to calculate grades. Well, I just did it. With this online grade calculator you are able to input your assignment/test scores and the calculator will tell your final score and final letter grade. And if you don't have all the grades, the calculator will tell you what score you need to get in your final assignment/test in order to get an A.

After some time of silence, I'm back. As many of you know, I run this site on my spare time trying to keep it as updated as possible. I'm very pleased to see the positive effect the math tools we offer in our site are having in the community. Countless e-mails thanking me for the work I put in the site make me want to keep working even harder. The free online math calculator we provide in the site have even been used as references in scientific papers. That is very cool indeed! I have great plans for the site. Keep visiting.

Our good friends at MGT are giving specially for us this 15% discount code for their online math homework help services: mthcrk1501 . From references given by users of our community, they are very reliable.

Summer is over, and time is back for some math. We have great plans for this year. Our aim is to offer the best quality free math help, for which reason we will continue adding more and more useful content. We also expect to materialize some very interesting partnerships. Stay tuned!

As promised, a couple of critical values calculators have been added: z-distribution critical values and t-distribution critical values

A t-distribution probability calculator was added. This is the first delivery of a series of scripts that will be added. Coming soon a couple of online calculators for critical values for the z-distribution and the t-distribution.

I just added an article about math problem solving techniques. See article

We have added more than 300 solved statistics problems

Due to a problem with a configuration file the scripts in the site were not working for a while, now they are working fine. Sorry for the inconvenience.

A good tutorial about the normal distribution was added. The normal distribution and the standard normal distribution

I added a good list of solved statistics problems.

I made a few adjustments to the scatterplot module, and added a new regression line tool, to complement the existing tool. Linear Regression Calculator

This new resource seems to be very useful, you may find it interesting to take a look Free Statistics Help

The site is back up online. For problems out of my control the site was down for a couple of weeks. We'll be back with new material. Thanks to everyone.

I have found a good collection of free solved math problems and math answers.

In case you have any suggestion, please do not hesitate to contact us .

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Demystifying Math 55

By anastasia yefremova.

Few undergraduate level classes have the distinction of nation-wide recognition that Harvard University’s Math 55 has. Officially comprised of Mathematics 55A “Studies in Algebra and Group Theory” and Mathematics 55B “Studies in Real and Complex Analysis,” it is technically an introductory level course. It is also a veritable legend among high schoolers and college students alike, renowned as — allegedly — the hardest undergraduate math class in the country. It has been mentioned in books and articles, has its own Wikipedia page, and has been the subject of countless social media posts and videos.

Most recently, Harvard junior Mahad Khan created a TikTok video dedicated to Math 55 that has received over 360,000 views to date. His is only one of many — his older brother created one, too — but it has the distinction of an insider’s perspective. “I thought it would be interesting if I cleared up the misconceptions about Math 55,” Khan said. While he hadn’t taken the course himself, he wanted to go beyond its reputation. “I wanted to get a real perspective by interviewing a former student and current course assistant.”

Over the years, perception of Math 55 has become based less on the reality of the course itself and more on a cumulative collection of lore and somewhat sensationalist rumors. It’s tempting to get swept up in the thrill of hearsay but while there might be kernels of truth to some of the stories, many of them are outdated and taken out of context. At the end of the day, however, Math 55 is a class like any other. Below, we take a stab at busting some of the more well known and persistent myths about the class. Or, at the very least, offering an extra layer of clarity. 

Myth #1: Math 55 is only for high school math geniuses

Most articles or mentions of Math 55 refer to it as filled with math competition champions and genius-level wunderkinds. The class is supposedly legendary among high school math prodigies, who hear terrifying stories about it in their computer camps and at the International Math Olympiad. There are even rumors of a special test students have to take before they are even allowed into Math 55. But while familiarity with proof-based mathematics is considered a plus for those interested in the course, there is no prerequisite for competition or research experience. 

In fact students whose only exposure to advanced math has been through olympiads and summer research programs can have a harder time adjusting. Their approach to the material tends to be understandably more solitary and that can be a disadvantage for the level of collaboration higher level mathematics require. “It has become a lot more open to people with different backgrounds,” said Professor Denis Auroux , who teaches Math 55,. “Our slogan is, if you’re reasonably good at math, you love it, and you have lots of time to devote to it, then Math 55 is completely fine for you.” 

Also, there is no extra test to get into the class.

Myth #2: Just take a graduate class, instead

Math 55 is hard. Whether you’re just 55-curious, or a past or present student in the class, this is something everyone agrees on. The course condenses four years of math into two semesters, after all. “For the first semester, you work on linear and abstract algebra with a bit of representation theory,” said sophomore math concentrator Dora Woodruff. “The second semester is real and complex analysis, and a little bit of algebraic topology. That’s almost the whole undergraduate curriculum.” Woodruff — incidentally, the student Khan interviewed — took Math 55 as a freshman and returned her second year as a course assistant. She is intimately familiar with the course’s difficulty level.

So why not just take an upper level undergraduate course to begin with or even one at a graduate level, if you’re really looking for a challenge? What justifies the existence of a class with the difficulty level of Math 55? One argument is that the course helps structure and systemize the knowledge with which many students come to Harvard. It gives them a firm background in preparation for the rest of their math education. Math 55 is difficult and it is purposefully structured that way as it’s meant to help students mature as mathematicians rather than as simple course takers.

But more importantly, “it’s just not true that Math 55 is at the level of a graduate class,” Auroux said. “It goes through several upper division undergraduate math classes with maybe a bit more advanced digressions into material here and there, but it sticks very close to what is taught in 100-level classes. The difference is we go through it at a faster pace, maybe with more challenging homework, and ideally as a community of people bringing our heads together.” 

A core goal of Math 55, according to Auroux, is to build a sense of community. Other schools might encourage advanced first-year students to take upper level undergraduate or even graduate classes, but Math 55 helps build a cohort of like-minded people who really like math, are good at it, and want to do a lot of it during their time at Harvard. That’s the experience Woodruff had, as well. “The community can be very strong,” she said. “You meet a lot of other people very interested in math and stay friends with them for the rest of college.”

Myth #3: Homework takes between 24 and 60 hours

Horror stories of endless homework are synonymous with the class. You’ll read or hear about “24 to 60 hours per week on homework” in almost every reference to Math 55. But one, there is a world of difference between 24 and 60 hours that is never explained, and two, this timeframe is quite misaligned with reality.

Auroux frequently sends out surveys to his students asking how long homework takes them and the average for most is closer to 15 hours a week. Those with more extensive prior math backgrounds can take as little as five to ten hours. The key factor is collaboration. “This class doesn’t lend itself to self-study,” Auroux stressed. Once they have thought about each problem set on their own, students are welcome and encouraged to talk to their friends and collaborate. “As soon as I see that something took over 30 hours I ask the student, do you know you’re supposed to be working with people and come ask me questions when you’re stuck?”

It is true that between reviewing lectures, digesting the material, and solving the problem sets, students usually end up devoting between 20 and 30 hours a week to the class. However, that includes the time dedicated to homework. So while students are discouraged from taking too many difficult classes and extracurriculars in the same semester as Math 55, they are also not expected to spend the time equivalent to a full-time job on their problem sets every week.

Myth #4: less than half of the class makes it to the second semester

Math 55 is just as infamous for its attrition rate as it is for its difficulty. Most sources like to cite the 1970 class, which began with 75 students and — between the advanced nature of the material and the time-constraints under which students had to work — ended with barely 20. Since then, the rumor has been that the Math 55 class shrinks by half its original size or more before the first semester is over. The reality is much less shocking and a bit more complicated.

Enrollment in this past fall semester’s Math 55A peaked at (ironically) 55 students. Well into the spring semester’s Math 55B, 47 students were still enrolled in the course. “On average, a drop of about 10-15 percent is much closer to what I would expect,” Auroux said. And those numbers become even more flexible if one takes into consideration the weeks math students have at the beginning of each semester to try out different classes and “shop” around before they have to commit to anything. This means students find their way in and out of Math 55 in a variety of ways over the course of the academic year.

According to Auroux, some students shop Math 55 in the fall and switch to the less intense Math 25 for the remainder of the semester. Others start out in Math 25 and, if not sufficiently challenged, switch to Math 55. Even people who end up in academia are not exempt from this. During his time as a student, our own Department of Mathematics’ Professor Emeritus Benedict Gross switched to the lower level Math 21 after two weeks in Math 55. In fact, those two weeks almost made him reconsider his desire to pursue mathematics. “By the beginning of sophomore year, I had decided to major in physics,” he recalled. “But during shopping period that fall, I walked past a math class taught by Andrew Gleason and stopped in to listen. It turned out to be Math 55.” He enrolled and by the end of the semester had found his vocation in mathematics.

All this means that Auroux sees student numbers vacillate up and down throughout the academic year. “There are about four or five students in this spring semester’s Math 55 that took Math 25 or even Math 22 in the fall, and they’re doing mostly fine,” he said. “It’s a lot of work, but I think they’re having a great time.”

Myth #5: 55-er culture is cult-y and exclusionary

Even though her experience with Math 55 was a positive one, Woodruff is very aware of the unhealthy culture the class has been rumored to cultivate. It’s easy for students to form exclusionary cliques that consist only of other Math 55 students, and some look down on anyone taking lower level math classes. But Woodruff also stressed that the instructors are very aware of this and actively take steps to curb that kind of toxic behavior. She said Auroux frequently brings up the importance of keeping the Math 55 community inclusive through Slack messages and lecture references.

Some students come to Harvard just for the opportunity to take Math 55. Some view enrolling in the class as proof of their mathematical gumption and competence. A Harvard Independent article called Math 55 the “premiere mathematical challenge for overachieving and…ridiculously mathy freshmen” and a piece in The Harvard Crimson referred to it as “a bit of a status thing as far as math majors here are concerned.” Over the years, the Harvard Department of Mathematics has taken steps to correct these assumptions. 

For one thing, neither the Math 55A nor the Math 55B official course descriptions boast the dubious honor of referring to it as “probably the most difficult undergraduate math class in the country” (don’t trust everything you read on Wikipedia). For another, “we’re trying to emphasize that there’s no magic to Math 55,” Auroux said. “It contains the same material as some of the other classes we have. People who take it are not intrinsically better or smarter than the ones who don’t.” 

Myth #6: You have to take Math 55 if you’re serious about going into academia

One reason math concentrators could feel pressured to enroll in Math 55 is because they view it as a prerequisite for a career in academia. It’s a sort of badge of honor and proof of their commitment to the field of mathematics. It is true that quite a few graduates of the course have gone on to pursue a career in mathematics. Woodruff herself believes that will be the most likely path for her, and several faculty members in our own Department of Mathematics took Math 55 during their days as Harvard freshmen.

“Several times in my research career when I understood something fundamental, I would realize that this was what Math 55 was trying to teach us,” Gross said. “It was an amazing introduction to the whole of mathematics and it was transformative for me.” In fact, Gross met Higgins Professor of Mathematics Joe Harris when they took the class together, forging a lifelong friendship. When they returned to Harvard as faculty, they took turns teaching Math 25 and Math 55. 

However, Auroux is quick to point out that while many graduates of the course do end up in academia, most professional mathematicians have likely never even heard of Math 55. “I would like to think that it’s a success story if people end up doing math, because the goal of Math 55 is to show students how beautiful math can be,” he said. “If they love it enough to go to grad school and become mathematicians, that’s wonderful. And if they want to take that math knowledge and do something else with their life, that’s just as wonderful.”

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• Differentiation: definition and basic derivative rules
• Differentiation: composite, implicit, and inverse functions
• Contextual applications of differentiation
• Applying derivatives to analyze functions
• Integration and accumulation of change
• Applications of integration
• AP Calculus AB solved free response questions from past exams
• AP®︎ Calculus AB Standards mappings
• Infinite sequences and series
• AP Calculus BC solved exams
• AP®︎ Calculus BC Standards mappings
• Integrals review
• Integration techniques
• Thinking about multivariable functions
• Derivatives of multivariable functions
• Applications of multivariable derivatives
• Integrating multivariable functions
• Green’s, Stokes’, and the divergence theorems
• First order differential equations
• Second order linear equations
• Laplace transform
• Vectors and spaces
• Matrix transformations
• Alternate coordinate systems (bases)

Frequently Asked Questions about Khan Academy and Math Worksheets

Why is khan academy even better than traditional math worksheets.

Khan Academy’s 100,000+ free practice questions give instant feedback, don’t need to be graded, and don’t require a printer.

What do Khan Academy’s interactive math worksheets look like?

Here’s an example:

What are teachers saying about Khan Academy’s interactive math worksheets?

“My students love Khan Academy because they can immediately learn from their mistakes, unlike traditional worksheets.”

Is Khan Academy free?

Khan Academy’s practice questions are 100% free—with no ads or subscriptions.

What do Khan Academy’s interactive math worksheets cover?

Our 100,000+ practice questions cover every math topic from arithmetic to calculus, as well as ELA, Science, Social Studies, and more.

Is Khan Academy a company?

Khan Academy is a nonprofit with a mission to provide a free, world-class education to anyone, anywhere.

Want to get even more out of Khan Academy?

Then be sure to check out our teacher tools . They’ll help you assign the perfect practice for each student from our full math curriculum and track your students’ progress across the year. Plus, they’re also 100% free — with no subscriptions and no ads.

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