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

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Exponents

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

Exponents to Home Page