# Exponents Formula

In the expression, $a^{2}$, a is known as base and 2 is known as the exponent. An exponent represents the number of times the base to be multiplied. For example, in $a^{2}$, a will be multiplied twice, i.e., a $\times$ a and silimarlt $a^{3}$ = a $\times$ a $\times$ a.

Here we will learn about various formulas of exponents

#### The Exponents Formulas are

$\large a^{0}=1$

$\large a^{1}=a$

$\large \sqrt{a}=a^{\frac{1}{2}}$

$\large \sqrt[n]{a}=a^{\frac{1}{n}}$

$\large a^{-n}=\frac{1}{a^{n}}$

$\large a^{n}=\frac{1}{a^{-n}}$

$\large a^{m}a^{n}=a^{m+n}$

$\large \frac{a^{m}}{a^{n}}=a^{m-n}$

$\large (a^{m})^{p}=a^{mp}$

$\large (a^{m}c^{n})^{p}=a^{mp}c^{np}$

$\large \left ( \frac{a^{m}}{c^{n}} \right )^{p}=\frac{a^{mp}}{c^{np}}$

### Solved Examples

Question 1: Solve $\frac{1}{4^{-3}}$

Solution: As per the The Negative Exponent Rule –

$\frac{1}{a^{-n}}=a^{n}$

$\frac{1}{4^{-3}} = 4^{3} = 64$

Question 2: Solve $\large\frac{3a^{-3}b^{5}}{4a^{4}b^{-3}}$

= $\large\frac{3b^{3}b^{5}}{4a^{4}a^{3}}$

= $\large\frac{3b^{8}}{4a^{7}}$

 More topics in Exponential Formula Square Root Formula Sum of Squares Formula Difference of Squares Formula Cube Formula Cube Root Formula Binomial Expansion Formula Exponential Function Formula Exponential Equation Formula Double Time Formula

#### Practise This Question

For the damped oscillator shown, the mass m of the block is 200 g, k = 90 N m1 and the damping constant b is 40 g s1. What is the period of oscillation, time taken for its amplitude of vibrations to drop to half of its initial value.