Exponents Formula

Exponents Formula

In the expression

\(\begin{array}{l}a^{2}\end{array} \)
, 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
\(\begin{array}{l}a^{2}\end{array} \)
, a will be multiplied twice, i.e., a 
\(\begin{array}{l}\times\end{array} \)
a and similarly,
\(\begin{array}{l}a^{3}\end{array} \)
= a
\(\begin{array}{l}\times\end{array} \)
\(\begin{array}{l}\times\end{array} \)

Here you will learn about various formulas of exponents.

The Exponents Formulas are

\(\begin{array}{l}\large a^{0}=1\end{array} \)
\(\begin{array}{l}\large a^{1}=a\end{array} \)
\(\begin{array}{l}\large \sqrt{a}=a^{\frac{1}{2}}\end{array} \)
\(\begin{array}{l}\large \sqrt[n]{a}=a^{\frac{1}{n}}\end{array} \)
\(\begin{array}{l}\large a^{-n}=\frac{1}{a^{n}}\end{array} \)
\(\begin{array}{l}\large a^{n}=\frac{1}{a^{-n}}\end{array} \)
\(\begin{array}{l}\large a^{m}a^{n}=a^{m+n}\end{array} \)
\(\begin{array}{l}\large \frac{a^{m}}{a^{n}}=a^{m-n}\end{array} \)
\(\begin{array}{l}\large (a^{m})^{p}=a^{mp}\end{array} \)
\(\begin{array}{l}\large (a^{m}c^{n})^{p}=a^{mp}c^{np}\end{array} \)
\(\begin{array}{l}\large \left ( \frac{a^{m}}{c^{n}} \right )^{p}=\frac{a^{mp}}{c^{np}}\end{array} \)


Solved Examples

Question 1: Solve:

\(\begin{array}{l}\frac{1}{4^{-3}}\end{array} \)

Solution: As per the The Negative Exponent Rule –

\(\begin{array}{l}\frac{1}{a^{-n}}=a^{n}\end{array} \)

\(\begin{array}{l}\frac{1}{4^{-3}} = 4^{3} = 64\end{array} \)

Question 2: Solve:

\(\begin{array}{l}\large\frac{3a^{-3}b^{5}}{4a^{4}b^{-3}}\end{array} \)


\(\begin{array}{l}\large\frac{3a^{-3}b^{5}}{4a^{4}b^{-3}}\end{array} \)


\(\begin{array}{l}\large\frac{3b^{3}b^{5}}{4a^{4}a^{3}}\end{array} \)


\(\begin{array}{l}\large\frac{3b^{8}}{4a^{7}}\end{array} \)

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


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  1. What is the formula for law of exponents