Principle 2.9. Power of a Power.
\begin{equation*}
\left(a^n\right)^m = (a^m)^n = a^{m n}
\end{equation*}
Section 2.3 Rational Exponents \(x^{m/n}\)
When taking a power of a power, you should multiply the exponents. For example,
\begin{equation*}
\left(2^2\right)^3 = 2^2\cdot 2^2\cdot 2^2 = 2^{3\cdot 2}=2^6.
\end{equation*}
This is the same as
\begin{equation*}
\left(2^3\right)^2 = 2^3\cdot 2^3 = 2^{2\cdot 3}=2^6.
\end{equation*}
What happens when we apply this law to an exponent that is a fraction like \(1/2\text{?}\) Consider,
\begin{equation*}
\left(2^{1/2}\right)^{2} = 2^{(1/2)\cdot 2} = 2^1 = 2.
\end{equation*}
Observe that \(2^{1/2}\) is a number whose square is \(2\text{,}\) just like \(\sqrt{2}\text{.}\) In fact, we define \(2^{1/2}=\sqrt{2}\text{.}\) In general, we define the reciprocal exponent as a principal \(n\)-th root.
Definition 2.10. Reciprocal Exponent.
For positive integer \(n\text{,}\) we define
\begin{equation*}
a^{1/n} = \sqrt[n]{a}.
\end{equation*}
We also assume \(a\geq 0\) in the case that \(n\) is even.
Example 2.11.
The cube root of \(8\) in exponential notation is
\begin{equation*}
8^{1/3} = \sqrt[3]{8} = 2.
\end{equation*}
The cube root of \(-8\) is
\begin{equation*}
(-8)^{1/3} = \sqrt[3]{-8} = -2.
\end{equation*}
Recall that a rational number is a ratio of integers \(m/n\text{,}\) where \(n\neq 0\text{.}\) We can now make sense of expressions with rational exponents.
Principle 2.12. Rational Exponent.
\begin{equation*}
a^{m/n} = \left(a^m\right)^{1/n} = \sqrt[n]{a^m}
\end{equation*}
Or equivalently,
\begin{equation*}
a^{m/n} = \left(a^{1/n}\right)^{m} = \left(\sqrt[n]{a}\right)^m.
\end{equation*}
Example 2.13.
To illustrate,
\begin{equation*}
4^{3/2} = (4^{1/2})^3 = (\sqrt{4})^3 = 2^3 = 8.
\end{equation*}
Alternatively,
\begin{equation*}
4^{3/2} = (4^3)^{1/2} = \sqrt{4^3} = \sqrt{64} = 8.
\end{equation*}
Notice both ways give the same result.
Example 2.14.
Evaluate each of the following expressions without using a calculator.
- \(\displaystyle \displaystyle -64^{3/2}\)
- \(\displaystyle \displaystyle 64^{-3/2}\)
- \(\displaystyle \displaystyle (-64)^{2/3}\)
- \(\displaystyle \displaystyle \frac{1}{{64}^{-2/3}}\)
Solution.
First, observe that \(8^2 = 64\) and \(4^3 = 64\text{.}\)
- Order of operations requires us to perform the exponent before multiplying by \(-1\)\begin{equation*} -64^{3/2} = - \left(\sqrt{64}\right)^3 = - 8^3 = -512. \end{equation*}
- Recall the negative exponents means reciprocal\begin{equation*} 64^{-3/2} = \frac{1}{64^{3/2}} = \frac{1}{\left(\sqrt{64}\right)^3} = \frac{1}{8^3} = \frac{1}{512} \end{equation*}
- Order of operations requires us to perform the operations inside parenthesis before the exponent.\begin{equation*} (-64)^{2/3} = \left(\sqrt[3]{-64}\right)^2 = \left(-4\right)^2 = 16 \end{equation*}
- The exponent is performed before the division.\begin{equation*} \frac{1}{{64}^{-2/3}} = 64^{2/3} = \left(\sqrt[3]{64}\right)^2 = \left(4\right)^2 = 16 \end{equation*}
