Laws of Exponents

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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) 33 x 3 2

= 3(3 + 2) = 35[exponents are added]

ii) b5 x b-2

= b5-2

= b3

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

= (-6)3+2

= (-6)5

(iv) 810 x 812

= 810+12

= 822

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) 45/ 43

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

= 4( 5 – 3)

= 42

(ii) p6÷p2 = p6 - 2
= p4
(iii) 815/812

= 815-12

= 83

(iv) 156/158

= 156-8

= 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) (23)2

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

= 26

(ii)(-84)2

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

= (-8)8

(iii) (y-2)-3

= y(-2 x -3)

= y6 [ 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) a0

= 1

(iii) (-25)0

= 1

Exponent 1

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

 a1 = a

Example :

(i) 201

= 20

(ii) b1

= 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) 22 x 32

= ( 2 x 3 )2

= 62

= 6 x 6 = 36

(ii) 62 ÷ 32

= ( 6/3)2

= 22

= 2 x 2 = 4

(iii) 34 x 3-3

= 34 ÷ 33

= 34 / 33

= 81 / 27

= 3

Exponents

Laws of Exponents
Rational Exponents
Integral Exponents
Scientific notation
Solved examples on Scientific Notation
Solved Examples on Exponents