Laws Of Exponents

 

Exponents are also called Powers or Indices

The exponent of a number says how many times to use the number in a multiplication.

In this example: 72 = 7 × 7 = 49

In words: 72 could be called "7 to the second power", "7 to the power 2" or simply "7 squared"

DEMONSTRATION:

 b7

b7 = b × b × b × b × b × b × b = bbbbbbb

Exponent rules

What is an exponent?

The base a raised to the power of n is equal to the multiplication of a, n times:

a n = × a × ... × a

n times

a is the base and n is the exponent.

Examples:

31 = 3

32 = 3 × 3 = 9

33 = 3 × 3 × 3 = 27

34 = 3 × 3 × 3 × 3 = 81

35 = 3 × 3 × 3 × 3 × 3 = 243

Exponents rules and properties

Rule name

Rule

Example

Product rules

a n ⋅ a m = a n+m

23 ⋅ 24 = 23+4 = 128

a n ⋅ b n = ( b) n

32 ⋅ 42 = (3⋅4)2 = 144

Quotient rules

a n / a m = a n-m

25 / 23 = 25-3 = 4

a n / b n = (/ b) n

43 / 23 = (4/2)3 = 8

Power rules

(bn)m = bnm

(23)2 = 232 = 64

bnm = b(nm)

232 = 2(32)= 512

m√(bn) = b n/m

2√(26) = 26/2 = 8

b1/n = nb

81/3 = 3√8 = 2

Negative exponents

b-n = 1 / bn

2-3 = 1/23 = 0.125

Zero rules

b0 = 1

50 = 1

0n = 0 , for n>0

05 = 0

One rules

b1 = b

51 = 5

1n = 1

15 = 1

Minus one rule

 

(-1)5 = -1

Derivative rule

(xn)nx n-1

(x3)= 3⋅x3-1

Integral rule

 xndx = xn+1/(n+1)+C

 x2dx = x2+1/(2+1)+C