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Section 13.8 Rational Exponents

The power to a power rule of exponents relates that \((x^m)^n=x^{mn}\text{.}\) This rule is fairly intuitive when both exponents are positive. For example, in the expression \((x^4)^3\) there are three factors of \(x^4\text{,}\) each of which contains four factors of \(x\text{,}\) so all together there are four factors of \(x\text{,}\) three times, i.e. \(3 \cdot 4\) factor of \(x\text{.}\)

While the power to a power rule is less intuitive once you move away from positive integer exponents, the rule remains the same regardless of the nature of the exponents. For example:

\begin{align*} (x^{1/3})^3\amp=x^{\frac{1}{3} \cdot 3}\\ \amp=x^{1}\\ \amp=x \end{align*}

But we already have a name for the expression that when cubed results in \(x\text{,}\) and that name is \(\sqrt[3]{x}\) (the cube root of \(x\)). So it must be the case that \(x^{1/3}=\sqrt[3]{x}\text{.}\) In general, is \(n\) is any positive integer, then:

\begin{equation*} x^{1/n}=\sqrt[n]{x} \end{equation*}

and more generally,

\begin{equation*} x^{m/n}=\sqrt[n]{x^m}\text{.} \end{equation*}

Several examples are shown below.

Example 13.8.1.

Express \(y^{7/5}\) as an equivalent radical expression

Solution
\begin{equation*} y^{7/5}=\sqrt[5]{y^7} \end{equation*}
Example 13.8.2.

Express \(\sqrt[3]{w^{12}}\) using an equivalent exponential expression

Solution
\begin{align*} \sqrt[3]{w^{12}}\amp=w^{12/3}\\ \amp=w^4 \end{align*}
Example 13.8.3.

Express \(\sqrt{x^9}\) using an equivalent exponential expression

Solution
\begin{equation*} \sqrt{x^9}=x^{9/2} \end{equation*}

You can use Figureย 13.8.4 to explore this definition some more.

Figure 13.8.4. Explore the Meaning of Rational Exponents

As long as both the numerator and denominator of a rational exponent are fairly small positive numbers, it is fairly easy to evaluate expressions that include rational exponents using the rule \(x^{m/n}=\sqrt[n]{x^m}\text{.}\)

Example 13.8.5.

Evaluate \(16^{1/2}\text{.}\)

Solution
\begin{align*} 16^{1/2}\amp=\sqrt{16}\\ \amp=4 \end{align*}
Example 13.8.6.

Evaluate \(8^{2/3}\text{.}\)

Solution
\begin{align*} 8^{2/3}\amp=\sqrt[3]{8^2}\\ \amp=\sqrt[3]{64}\\ \amp=4 \end{align*}
Example 13.8.7.

Evaluate \(100^{3/2}\text{.}\)

Solution
\begin{align*} 100^{3/2}\amp=\sqrt{100^3}\\ \amp=\sqrt{1000000}\\ \amp=1000 \end{align*}

When the numerator of the rational exponent is large, the rule \(x^{m/n}=\sqrt[n]{x^m}\) can become quite cumbersome. Consider, for example, evaluating \(9^{5/2}\text{.}\) If we try to use the standard form we hit a brick wall. First, it's not trivial to calculate that \(9^5=59,049\) (reality check ... I grabbed my calculator). Now that I have the value of 59,049, I have to determine its square root. Oh my!

Fortunately for us, the application of the exponent and the application of the radical can be done in either order. That is:

\begin{equation*} a^{m/n}=\sqrt[n]{x^m} \text{ and } a^{m/n}=(\sqrt[n]{x})^m \end{equation*}
Example 13.8.8.

Using the second option, evaluate \(9^{5/2}\text{.}\)

Solution
\begin{align*} 9^{5/2}\amp=(\sqrt{9})^5\\ \amp=3^5\\ \amp=243 \end{align*}
Example 13.8.9.

Using the second option, evaluate \(16^{7/4}\text{.}\)

Solution
\begin{align*} 16^{7/4}\amp=(\sqrt[4]{16})^7\\ \amp=2^7\\ \amp=128 \end{align*}

Rational exponents are allowed to be negative. If that's the case, you probably want to deal with the negative aspect of the exponent before taking on the fractional aspect.

Example 13.8.10.

Evaluate \(27^{-2/3}\text{.}\)

Solution
\begin{align*} 27^{-2/3}\amp=\frac{1}{27^{2/3}}\\ \amp=\frac{1}{(\sqrt[3]{27})^2}\\ \amp=\frac{1}{3^2}\\ \amp=\frac{1}{9} \end{align*}

Sometimes radical expressions can be simplified after first rewriting the expressions using rational exponents and applying the appropriate rules of exponents. If the resultant expression still has a rational exponent, it is standard to convert back to radical notation. Several examples follow.

Example 13.8.11.

Use rational exponents to simplify \(\text{.}\) Where appropriate, your final result should be converted back to radical form.

Solution
\begin{align*} \sqrt[3]{y^2} \cdot \sqrt[6]{y}\amp=y^{2/3}y^{1/6}\\ \amp=y^{2/3+1/6}\\ \amp=y^{5/6}\\ \amp=\sqrt[6]{y^5} \end{align*}
Example 13.8.12.

Use rational exponents to simplify \(\sqrt[8]{t^4}\text{.}\) Where appropriate, your final result should be converted back to radical form.

Solution
\begin{align*} \sqrt[8]{t^4}\amp=t^{4/8}\\ \amp=t^{1/2}\\ \amp=\sqrt{t} \end{align*}
Example 13.8.13.

Use rational exponents to simplify \(\sqrt[10]{\sqrt{5^{40}}}\text{.}\) Where appropriate, your final result should be converted back to radical form.

Solution
\begin{align*} \sqrt[10]{\sqrt{5^{40}}}\amp=\sqrt[10]{5^{40/2}}\\ \amp=\sqrt[10]{5^{20}}\\ \amp=5^{20/10}\\ \amp=5^2\\ \amp=25 \end{align*}

Exercises Exercises

Convert each exponential expression to a radical expression and each radical expression to an exponential expression. When converting to a rational exponent, reduce the exponent if possible. Assume that all variables represent positive values.

1.

\(x^{1/3}\)

Solution

\(x^{1/3}=\sqrt[3]{x}\)

2.

\(y^{5/4}\)

Solution

\(y^{5/4}=\sqrt[4]{y^5}\)

3.

\(z^{2/5}\)

Solution

\(z^{2/5}=\sqrt[5]{z^2}\)

4.

\(\sqrt[11]{x^5}\)

Solution

\(\sqrt[11]{x^5}=x^{5/11}\)

5.

\(\sqrt[4]{y^{20}}\)

Solution

\(\begin{aligned}[t] \sqrt[4]{y^{20}}\amp=y^{20/4}\\ \amp=y^5 \end{aligned}\)

6.

\(\sqrt[15]{t^3}\)

Solution

\(\begin{aligned}[t] \sqrt[15]{t^3}\amp=t^{3/15}\\ \amp=t^{1/5} \end{aligned}\)

Determine the value of each expression.

7.

\(4^{1/2}\)

Solution

\(\begin{aligned}[t] 4^{1/2}\amp=\sqrt{4}\\ \amp=2 \end{aligned}\)

8.

\(27^{-1/3}\)

Solution

\(\begin{aligned}[t] 27^{-1/3}\amp=\frac{1}{27^{1/3}}\\ \amp=\frac{1}{\sqrt[3]{27}}\\ \amp=\frac{1}{3} \end{aligned}\)

9.

\(\left(\frac{4}{9}\right)^{-1/2}\)

Solution

\(\begin{aligned}[t] \left(\frac{4}{9}\right)^{-1/2}\amp=\left(\frac{9}{4}\right)^{1/2}\\ \amp=\sqrt{\frac{9}{4}}\\ \amp=\frac{3}{2} \end{aligned}\)

10.

\(8^{7/3}\)

Solution

\(\begin{aligned}[t] 8^{7/3}\amp=(\sqrt[3]{8})^7\\ \amp=2^7\\ \amp=128 \end{aligned}\)

11.

\(100^{5/2}\)

Solution

\(\begin{aligned}[t] 100^{5/2}\amp=(\sqrt{100})^5\\ \amp=10^5\\ \amp=100,000 \end{aligned}\)

12.

\(16^{-9/4}\)

Solution

\(\begin{aligned}[t] 16^{-9/4}\amp=\frac{1}{16^{9/4}}\\ \amp=\frac{1}{(\sqrt[4]{16})^9}\\ \amp=\frac{1}{2^9}\\ \amp=\frac{1}{512} \end{aligned}\)

Simplify each radical expression after first rewriting the expression in exponential form. Assume that all variables represent positive values. Where appropriate, your final result should be converted back to radical form.

13.

\(\sqrt[5]{t^{20}}\)

Solution

\(\begin{aligned}[t] \sqrt[5]{t^{20}}\amp=t^{20/5}\\ \amp=t^4 \end{aligned}\)

14.

\(6\sqrt[33]{x^{77}}\)

Solution

\(\begin{aligned}[t] 6\sqrt[33]{x^{77}}\amp=6x^{77/33}\\ \amp=6x^{7/3}\\ \amp=6\sqrt[3]{x^7} \end{aligned}\)

15.

\((\sqrt{3})^{10}\)

Solution

\(\begin{aligned}[t] (\sqrt{3})^{10}\amp=3^{10/2}\\ \amp=3^5\\ \amp=243 \end{aligned}\)

16.

\(\sqrt[4]{9^2}\)

Solution

\(\begin{aligned}[t] \sqrt[4]{9^2}\amp=9^{2/4}\\ \amp=9^{1/2}\\ \amp=\sqrt{9}\\ \amp=3 \end{aligned}\)

17.

\(\sqrt{w}\sqrt[4]{w}\)

Solution

\(\begin{aligned}[t] \sqrt{w}\sqrt[4]{w}\amp=w^{1/2}w^{1/4}\\ \amp=w^{3/4}\\ \amp=\sqrt[4]{w^3} \end{aligned}\)

18.

\(\sqrt[7]{x^6}\sqrt[7]{x}\)

Solution

\(\begin{aligned}[t] \sqrt[7]{x^6}\sqrt[7]{x}\amp=x^{6/7}x^{1/7}\\ \amp=x^1\\ \amp=x \end{aligned}\)

19.

\((\sqrt[12]{x^7y^{16}})^{36}\)

Solution

\(\begin{aligned}[t] (\sqrt[12]{x^7y^{15}})^{36}\amp=(x^7y^{16})^{36/12}\\ \amp=(x^7y^{16})^3\\ \amp=x^{21}y^{48} \end{aligned}\)

20.

\(\sqrt[5]{\sqrt[3]{x^{15}}}\)

Solution

\(\begin{aligned}[t] \sqrt[15]{\sqrt[3]{x^{15}}}\amp=\sqrt[15]{x^{15/3}}\\ \amp=\sqrt[15]{x^5}\\ \amp=x^{5/15}\\ \amp=x^{1/3}\\ \amp=\sqrt[3]{x} \end{aligned}\)