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.

\begin{array}{l} a^{0} = 1 \end{array}

\begin{array}{l} a^{1} = a \end{array}

\begin{array}{l} \sqrt{a} = a^{\frac{1}{2}} \end{array}

\begin{array}{l} \sqrt[n]{a} = a^{\frac{1}{n}} \end{array}

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

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

\begin{array}{l} a^{m} a^{n} = a^{m + n} \end{array}

\begin{array}{l} \frac{a^{m}}{a^{n}} = a^{m - n} \end{array}

\begin{array}{l} (a^{m})^{p} = a^{m p} \end{array}

\begin{array}{l} (a^{m} c^{n})^{p} = a^{m p} c^{n p} \end{array}

\begin{array}{l} {(\frac{a^{m}}{c^{n}})}^{p} = \frac{a^{m p}}{c^{n p}} \end{array}

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} \frac{3 a^{- 3} b^{5}}{4 a^{4} b^{- 3}} \end{array}

Solution:

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

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

\begin{array}{l} \frac{3 b^{8}}{4 a^{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