Exponent rules, laws of exponent and examples.

### What is an exponent

The base a raised to the power of n is equal to the multiplication of a, n times:

*a ^{n} = a × a × … × a* (n times)

a is the base and n is the exponent.

**Examples**

3^{1} = 3

3^{2} = 3 × 3 = 9

3^{3} = 3 × 3 × 3 = 27

3^{4} = 3 × 3 × 3 × 3 = 81

3^{5} = 3 × 3 × 3 × 3 × 3 = 243

3^{10} = 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 × 3 = 59 049

### Exponents rules and properties

Rule name | Rule | Example |

Product rules | a ^{n} ⋅ a ^{m} = a ^{n+m} | 2^{3} ⋅ 2^{4} = 2^{3+4} = 128 |

a ^{n} ⋅ b ^{n} = (a ⋅ b) ^{n} | 3^{2} ⋅ 4^{2} = (3⋅4)^{2} = 144 | |

Quotient rules | a ^{n} / a ^{m} = a ^{n-m} | 2^{5} / 2^{3} = 2^{5-3} = 4 |

a ^{n} / b ^{n} = (a / b) ^{n} | 4^{3} / 2^{3} = (4/2)^{3} = 8 | |

Power rules | (b^{n})^{m} = b^{n⋅m} | (2^{3})^{2} = 2^{3⋅2} = 64 |

b^{nm} = b^{(nm)} | 2^{32} = 2^{(32)} = 512 | |

^{m}√(b^{n}) = b ^{n/m} | ^{2}√(2^{6}) = 2^{6/2} = 8 | |

b^{1/n} = ^{n}√b | 8^{1/3} = ^{3}√8 = 2 | |

Negative exponents | b^{-n} = 1 / b^{n} | 2^{-3} = 1/2^{3} = 0.125 |

Zero rules | b^{0} = 1 | 5^{0} = 1 |

0^{n} = 0 , for n>0 | 0^{5} = 0 | |

One rules | b^{1} = b | 5^{1} = 5 |

1^{n} = 1 | 1^{5} = 1 | |

Minus one rule | (-1)^{5} = -1 | |

Derivative rule | (x^{n})' = n⋅x ^{n-1} | (x^{3})' = 3⋅x^{3-1} |

Integral rule | ∫ x^{n}dx = x^{n+1}/(n+1)+C | ∫ x^{2}dx = x^{2+1}/(2+1)+C |

### Exponents product rules

**Product rule with same base**

*a ^{n} ⋅ a^{m} = a^{n+m}*

Example:

2^{2} ⋅ 2^{3} = 2^{2+3} = 2^{5} = 2⋅2⋅2⋅2⋅2 = 32

**Product rule with same exponent**

*a ^{n} ⋅ b^{n} = (a ⋅ b)^{n}*

Example:

2^{2} ⋅ 3^{2} = (2⋅3)^{2} = 6^{2} = 6⋅6 = 36

### Exponents quotient rules

**Quotient rule with same base**

*a ^{n} / a^{m} = a^{n-m}*

Example:

2^{4} / 2^{2} = 2^{4-2} = 2^{2} = 2⋅2 = 4

**Quotient rule with same exponent**

*a ^{n} / b^{n} = (a / b)^{n}*

Example:

6^{3} / 3^{3} = (6/3)^{3} = 2^{3} = 2⋅2⋅2 = 8

### Exponents power rules

**Power rule I**

*(a ^{n}) ^{m} = a ^{n⋅m}*

Example:

(2^{1})^{3} = 2^{1⋅3} = 2^{3} = 2⋅2⋅2 = 8

**Power rule II**

*a ^{nm} = a ^{(nm)}*

Example:

2^{32} = 2^{(32)} = 2^{(3⋅3)} = 2^{9} = 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2 = 512

**Power rule with radicals**

^{m}√(a^{ n}) = a^{ n}^{/m}

Example:

^{2}√(2^{6}) = 2^{6/2} = 2^{3} = 2⋅2⋅2 = 8

### Negative exponents rule

*b ^{-n} = 1 / b^{n}*

Example:

2^{-3} = 1/2^{3} = 1/(2⋅2⋅2) = 1/8 = 0.125