# 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.

a^{m} x a^{n } = 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.

a^{m}÷ a^{n } = 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.

( a^{m}) ^{n} = a^{(m x n )} = a^{mn} |

**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.

a^{1} = 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/a^{m} |

1/a^{-m} = a^{m} |

**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.

a^{m} x b^{m} = (ab)^{m} |
a^{m} ÷ b^{m} = (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

• Laws of Exponents

• Rational Exponents

• Integral Exponents

• Scientific notation

• Solved examples on Scientific Notation

• Solved Examples on Exponents