# 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

= 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