# Laws of Exponents

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

There are 8 Laws of Exponents.

1) If the bases are same and there is a multiplication between them then, add the exponents keeping the base common.

i) 3

= 3

ii) b

= b

= b

= b

(iii) (-6)

= (-6)

= (-6)

(iv) 8

= 8

= 8

If the bases are same and there is a division between them then, subtract the 2nd exponent from the 1st keeping the base common.

(i) 4

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

= 4

= 4

(ii) p

= p

(iii) 8

= 8

= 8

(iv) 15

= 15

= 15

(v)(5/2)

= (5/2)

= (5/2)

3) If there are double exponents then, multiply the exponents and keep the base same.

(i) (2

= 2

= 2

(ii)(-8

= (-8)

= (-8)

(iii) (y

= y

= y

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

(i) (1000)

= 1

(ii) a

= 1

(iii) (-25)

= 1

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

(i) 20

= 20

(ii) b

= b

(iii) (2000)

= 2000

6) If the exponent is negative so to make it positive write the reciprocal of it.

i) 4

= 1 / 4

= 1 / 16

2) 1 / 3

= 3

7) Two different bases have same exponents then bring the two bases under common parenthesis and keep the same exponent.

(i) 2

= ( 2 x 3 )

= 6

= 6 x 6 = 36

(ii) 6

= ( 6/3)

= 2

= 2 x 2 = 4

(iii) 3

= 3

= 3

= 81 / 27

= 3

Exponents to Home Page

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.

**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.

**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.

**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.

**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.

**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.

**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.

**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

• Laws of Exponents

• Rational Exponents

• Integral Exponents

• Scientific notation

• Solved examples on Scientific Notation

• Solved Examples on Exponents