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 –


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

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

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

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

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.