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) 3³ x 3 ²

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

ii) b⁵ x b^-2

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

= b^5-2

= 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 = p^{6 - 2}
= p⁴
(iii) 8¹⁵/8¹²

= 8^15-12

= 8³

(iv) 15⁶/15⁸

= 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) (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

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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