Exponents are used to show repeated multiplication of a number by itself. Writing large numbers sometimes becomes tedious. In large mathematical expressions, they occupy more space and take more time. This issue is resolved by the use of exponents. For example, 7 × 7 × 7 can be represented as \( 7^3 \). In this example, the exponent is ‘3’ which stands for the number of times the value is multiplied by itself. The number 7 is called the base which is the actual number that is getting multiplied. For instance the speed of light is 300000000 m/s. This can be simply written as 3 × \( 10^8 \) m/s (approximate value). This process of using exponents is called as ‘raising to a power’ where exponent is the power.

## Exponents and Powers

We know that the expression 6 x 6 can be calculated, but the expression can also be written in a short manner that is known as exponents.

6.6 = 6^2

The expression that describes repetitive multiplication of same value is known as power. The value 6 is known as base or power and the number 2 is known as exponent. It corresponds to the number of times the base is operated as a factor.

### Exponent Rules

Rule name | Rule | Example |

Product rules | a n · a m = a n+m | 23 · 24 = 23+4 = 128 |

a n · b n = (a · b) n | 32 · 42 = (3·4)2 = 144 | |

Quotient rules | a n / a m = a n–m | 25 / 23 = 25-3 = 4 |

a n / b n = (a / b) n | 43 / 23 = (4/2)3 = 8 | |

Power rules | (bn)m = bn·m | (23)2 = 23·2 = 64 |

bnm = b(nm) | 232 = 2(32)= 512 | |

m√(bn) = b n/m | 2√(26) = 26/2 = 8 | |

b1/n = n√b | 81/3 = 3√8 = 2 | |

Negative exponents | b-n = 1 / bn | 2-3 = 1/23 = 0.125 |

Zero rules | b0 = 1 | 60 = 1 |

0n = 0 , for n>0 | 06 = 0 | |

One rules | b1 = b | 71 = 7 |

1n = 1 | 18 = 1 | |

Minus one rule | (-1)5 = -1 | |

Derivative rule | (xn)‘ = n·x n-1 | (x3)‘ = 3·x3-1 |

Integral rule | ∫ xndx = xn+1/(n+1)+C | ∫ x2dx = x2+1/(2+1)+C |

#### Power with positive and negative exponents

We know that \( 20^2 \)= 20 × 20 = 400

=> \( 20^1 \) = \( \frac {400}{20}\) = 20

=> \( 20^0 \) = \( \frac {20}{20}\) = 1

So,\( 20^{-1} \) = \( \frac {1}{20}\)

Similarly, \( 20^{-2} \) = \( \frac {1}{20}\) ÷ 20 = \( \frac {1}{20}\) × \( \frac {1}{20}\) = \( \frac {1}{20^2}\)

\( 20^{-3} = \frac {1}{20^3}\)

In general we can say that for any non-zero integer say ‘a’ , \( a^{-3} = \frac{1}{a^m} \) , where m is the positive integer.\( a^{- m} \) is also the multiplicative inverse of \( a^m\).

Illustration 1: Find the multiplicative inverse of \( 9^{-4} \)

Solution: \( 9^{-4} \) = \( \frac {1}{9^4} \)

Therefore the multiplicative inverse of \( \frac {1}{9^4} \) is \( 9^{4} \).

Illustration 2: Find the multiplicative inverse of \( 7^{2} \).

Solution: \( 7^{2} \) = \( \frac {1}{7^{-2}} \)

The multiplicative inverse of \( 7^{2} \) is \( 7^{-2} \).

Illustration 3: Expand the number 12345 in the exponent form.

Solution: The number 12345 can be expressed as:

12345 = 1 × 10000 + 2 × 1000 + 3 × 100 + 4 × 10 + 5 × 1

=> 12345 = 1 × \(10^4 ~+~ 2 ~×~ 10^3~ +~ 3~ ×~ 10^2 ~+~4~ ×~ 10^1~ +~ 5~ ×~ 10^0\) (any number raised to the power 0 is equal to 1).

Similarly this method can be employed to decimal numbers also.

Illustration 4: Expand the number 987.65 in the exponent form.

Solution: The number 987.65 can be represented as:

\(9~ ×~ 10^2~ +~ 8~ ×~ 10^1~ +~ 7~ ×~ 10^0~ +~ 6~ ×~ 10^{-1}~ +~ 5~ ×~ 10^{-2}\)