Laws of Exponents

Laws of Exponents includes laws of multiplication, division, double exponents,zero exponent etc.

There are 8 Laws of Exponents.
Multiplying powers with same base

1) If the bases are same and there is a multiplication between them then, add the exponents keeping the base common.
am x an = a ( m + n )

Examples :

i) 3 3 x 3 2

= 3 (3 + 2) = 3 5 [exponents are added]

ii) b 5 x b -2

= b 5 +(-2) [exponents are added]

= b 5-2

= b 3

(iii) (-6) 3 x (-6) 2

= (-6) 3+2

= (-6) 5

(iv) 8 10 x 8 12

= 8 10+12

= 8 22

Dividing powers with the same base

If the bases are same and there is a division between them then, subtract the 2nd exponent from the 1st keeping the base common.
am÷ an = a ( m - n )

Examples :

(i) 4 5 / 4 3

= (4 x 4 x 4 x 4 x 4)/(4 x 4 x 4)

= 4 ( 5 – 3)

= 4 2

(ii) p 6 ÷p 2 = p 6 - 2
= p 4
(iii) 8 15 /8 12

= 8 15-12

= 8 3

(iv) 15 6 /15 8

= 15 6-8

= 15 -2

(v)(5/2) 9 ÷ (5/2) 4

= (5/2) 9-4

= (5/2) 5

Power of a power

3) If there are double exponents then, multiply the exponents and keep the base same.
( am) n = a(m x n ) = amn

Examples :

(i) (2 3 ) 2

= 2 ( 3 x 2 ) [ multiply the two powers]

= 2 6

(ii)(-8 4 ) 2

= (-8) (4 x 2) [multiply the two powers]

= (-8) 8

(iii) (y -2 ) -3

= y (-2 x -3)

= y 6 [ negative times negative --->positive]

 Zero Exponent

4) Any number with exponent zero ,the answer is 1.
a 0 = 1

Example :

(i) (1000) 0

= 1

(ii) a 0

= 1

(iii) (-25) 0

= 1

Exponent 1

5) If the exponent is 1 then the number itself is the answer.
a1 = a

Example :

(i) 20 1

= 20

(ii) b 1

= b

(iii) (2000) 1

= 2000

Negative Exponent

6) If the exponent is negative so to make it positive write the reciprocal of it.
a-m = 1/am
1/a-m = am

Example :

i) 4 -2

= 1 / 4 2

= 1 / 16

2) 1 / 3 -2

= 3 2


7) Two different bases have same exponents then bring the two bases under common parenthesis and keep the same exponent.
am x bm = (ab)m am ÷ bm = (a/ b)m

Example 1 :

(i) 2 2 x 3 2

= ( 2 x 3 ) 2

= 6 2

= 6 x 6 = 36

(ii) 6 2 ÷ 3 2

= ( 6/3) 2

= 2 2

= 2 x 2 = 4

(iii) 3 4 x 3 -3

= 3 4 ÷ 3 3

= 3 4 / 3 3

= 81 / 27

= 3
Exponents

• Laws of Exponents
• Rational Exponents
• Integral Exponents
• Scientific notation
• Solved examples on Scientific Notation
• Solved Examples on Exponents
Exponents to Home Page