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.
\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}
Simplify $243^{3/5}$.
\[243^{3/5} = 243^{(3 \times 1/5)} = \left( 243^{1/5} \right)^3 = \left( \sqrt[5]{\strut 243} \right)^3 = 3^3=27.\]
Simplify $ 8^{-2/3} $.
\[8^{-2/3} = \frac{1}{ 8^{2/3} } = \frac{1}{(8^{1/3})^2} = \frac{1}{(\sqrt[3]{8})^2} = \frac{1}{2^2} = \frac{1}{4}\]
Simplify $ \left(\dfrac{81}{16}\right)^{-3/4} $.
\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}
Write the following expression in the form $a^n$ for some value $n$. \[\frac{a^2 \times a^5}{(a^3)^3}\]
\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}
Write the following expression in the form $a^n$ for some value of $n$. \[\sqrt{a} \times \frac{1}{ a^{-2} }.\]
\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}
Dr Jim Ford derives the laws of powers and gives examples of each.
Dr Jim Ford derives the laws of powers and gives examples of each.
Prof. Robin Johnson simplifies the expression $\dfrac{x^3y^2}{\sqrt{x^4/y}}$.