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Powers

Definition

Exponentiation is a mathematical operation, also known as “raising to a power”. The expression “$x$ raised to the power $y$” is written

\[x^y\]

$x$ is called the base of the operation and $y$ the exponent, power, or index.

When $y$ is a positive whole number, this can be interpreted as “$x$ multiplied by itself $y$ times”. This concept can be generalised to negative and fractional powers - these powers are defined to be consistent with the rules obtained from whole number powers.

Laws of indices

\begin{align} a^0 &= 1 \\ a^1 &= a \\\\ a^m \times a^n &= a^{m+n} \\\\ \dfrac{a^m}{a^n} &= a^{m-n} \\\\ (a^m)^n = a^{mn} &= (a^n)^m \\\\ a^n \times b^n &= (ab)^n \\\\ a^{-m} &= \dfrac{1}{a^m} \\\\ a^{1/n} &= \sqrt[n]{a} \end{align}

Worked Examples

Example 1

Simplify $243^{3/5}$.

Solution

\[243^{3/5} = 243^{(3 \times 1/5)} = \left( 243^{1/5} \right)^3 = \left( \sqrt[5]{\strut 243} \right)^3 = 3^3=27.\]

Example 2

Simplify $ 8^{-2/3} $.

Solution

\[8^{-2/3} = \frac{1}{ 8^{2/3} } = \frac{1}{(8^{1/3})^2} = \frac{1}{(\sqrt[3]{8})^2} = \frac{1}{2^2} = \frac{1}{4}\]

Example 3

Simplify $ \left(\dfrac{81}{16}\right)^{-3/4} $.

Solution

\begin{align} \left(\frac{81}{16}\right)^{-3/4} &= \frac{1}{ \left(\frac{81}{16}\right)^{3/4} }\\ &= \left(\frac{16}{81}\right)^{3/4}\\ &= \Biggl(\left(\frac{16}{81}\right)^{1/4}\Biggr)^3\\ &=\left(\frac{2}{3}\right)^3\\ &=\frac{8}{27} \end{align}

Example 4

Write the following expression in the form $a^n$ for some value $n$. \[\frac{a^2 \times a^5}{(a^3)^3}\]

Solution

\begin{align} \frac{a^2 \times a^5}{(a^3)^3} &= \frac{ a^{2+5} }{ a^{3\times 3} }\\ &=\frac{a^7}{a^9}\\ &=a^{7-9}\\ &=a^{-2} \end{align}

Example 5

Write the following expression in the form $a^n$ for some value of $n$. \[\sqrt{a} \times \frac{1}{ a^{-2} }.\]

Solution

\begin{align} \sqrt{a} \times \frac{1}{ a^{-2} } &= a^{ \frac{1}{2} } \times a^2\\ &= a^{\frac{1}{2}+2}\\ &= a^{5/2} \end{align}

Video Examples

Example 1

Dr Jim Ford derives the laws of powers and gives examples of each.

Example 2

Dr Jim Ford derives the laws of powers and gives examples of each.

Example 3

Prof. Robin Johnson simplifies the expression $\dfrac{x^3y^2}{\sqrt{x^4/y}}$.

Test Yourself

Powers and indices

External Resources

Whiteboard maths

More Support

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