To calculate the cube root of a number, say \(8\text{,}\) we can press the following buttons of a calculator:
\begin{equation*} 8\wedge(1/3) \end{equation*}This implies
\begin{equation*} \sqrt[3]{8}=8^{\frac{1}{3}} \end{equation*}In this section, we will learn why this is true.
Compare the following calculations:
\begin{align*} \sqrt{9}\cdot\sqrt{9}\amp=3\cdot3=9\\ 9^{\frac{1}{2}}\cdot9^{\frac{1}{2}}\amp=9^{\frac{1}{2}+\frac{1}{2}}=9^1=9 \end{align*}If we rewrite the above calculations with exponent, we have:
\begin{align*} (\sqrt{9})^2\amp=9\\ (9^{\frac{1}{2}})^2\amp=9 \end{align*}Since the square of \(\sqrt{9}\) and the square of \(9^{\frac{1}{2}}\) generates the same number, we conclude that
\begin{equation*} \sqrt{9}=9^{\frac{1}{2}} \end{equation*}We can verify this result by entering \(9\wedge(1/2)\) into a calculator, and the result should be \(3\text{.}\) In general, we have:
\begin{equation*} \sqrt{x}=x^{\frac{1}{2}} \end{equation*}Similarly, we can prove:
\begin{align*} \sqrt[3]{x}\amp=x^{\frac{1}{3}}\\ \sqrt[4]{x}\amp=x^{\frac{1}{4}}\\ \sqrt[5]{x}\amp=x^{\frac{1}{5}}\\ \ldots\\ \sqrt[n]{x}\amp=x^{\frac{1}{n}} \end{align*}With this relationship, we can re-write radical expressions into expressions with rational exponents.
Evaluate \(\sqrt[4]{9}\) with a calculator. Round your answer to two decimal places.
Since \(\sqrt[4]{9}=9^{\frac{1}{4}}\text{,}\) we press the following buttons on a calculator to get the value:
\begin{equation*} 9\wedge(1/4)\approx1.73\text{.} \end{equation*}In the next few examples, we need to use another important skill: recognizing the following special numbers:
| Special Numbers | |
| Perfect Square Numbers | \(0,1,4,9,16,25,36,49,64,81,100,121,144\) |
| Perfect Cube Numbers | \(0,1,8,27,64,125\) |
| Perfect \(4^{th}\)-Power Numbers | \(0,1,16,81\) |
| Powers of \(2\) | \(2,4,8,16,32,64,128,256,512,1024\) |
When we see one of these numbers in a problem, we must re-write the number into its prime factors. Most of the time, we will benefit from this conversion. Let's look at the next example.
Convert the following radical expressions into expressions with rational exponents, and simplify them if possible.
\(\sqrt[3]{5}\)
\(\sqrt[4]{4}\)
Solutions:
\(\sqrt[3]{5}=5^{\frac{1}{3}}\)
The second problem is more complicated because \(4\) is a special number:
\begin{align*} \sqrt[4]{4}\amp=4^{\frac{1}{4}}\\ \amp=(2^2)^{\frac{1}{4}}\\ \amp=2^{2\cdot\frac{1}{4}}\amp\text{because }(a^m)^n=a^{mn}\\ \amp=2^{\frac{1}{2}} \end{align*}Let's find a pattern and summarize an important property of radicals:
\begin{align*} \sqrt[3]{-1}\amp=-1\\ -\sqrt[3]{1}\amp=-1\\ \sqrt[3]{-8}\amp=-2\\ -\sqrt[3]{8}\amp=-2\\ \sqrt[3]{-27}\amp=-3\\ -\sqrt[3]{27}\amp=-3 \end{align*}In general, we have
\begin{equation*} \sqrt[3]{-a}=-\sqrt[3]{a} \end{equation*}Similarly, we can show
\begin{equation*} \sqrt[n]{-a}=-\sqrt[n]{a}\text{ where }n\text{ is odd} \end{equation*}Since \(\sqrt[n]{x}=x^{\frac{1}{n}}\text{,}\) the same property can be written as
\begin{equation*} (-a)^{\frac{1}{n}}=-a^{\frac{1}{n}}\text{ where }n\text{ is odd} \end{equation*}The next example uses this property.
Convert the following radical expressions into expressions with rational exponents, and simplify them if possible.
\(\sqrt[3]{4}\)
\(\sqrt[3]{-4}\)
Solution:
\begin{align*} \sqrt[3]{4}\amp=4^{\frac{1}{3}}\\ \amp=(2^2)^{\frac{1}{3}}\\ \amp=2^{2\cdot\frac{1}{3}}\\ \amp=2^{\frac{2}{3}} \end{align*}Solution:
\begin{align*} \sqrt[3]{-4}\amp=-\sqrt[3]{4}\\ \amp=-4^{\frac{1}{3}}\\ \amp=-(2^2)^{\frac{1}{3}}\\ \amp=-2^{2\cdot\frac{1}{3}}\\ \amp=-2^{\frac{2}{3}} \end{align*}The following example uses the property \(x^{-n}=\frac{1}{x^n}\text{.}\)
Convert the following radical expressions into expressions with rational exponents, and simplify them if possible.
\(\frac{1}{\sqrt{x}}\)
\(\frac{1}{\sqrt[3]{25}}\)
Solution:
\begin{align*} \frac{1}{\sqrt{x}}\amp=\frac{1}{x^{\frac{1}{2}}}\\ \amp=x^{-\frac{1}{2}}\amp\text{because }x^{-n}=\frac{1}{x^n} \end{align*}Solution:
\begin{align*} \frac{1}{\sqrt[3]{25}}\amp=\frac{1}{25^{\frac{1}{3}}}\\ \amp=\frac{1}{(5^2)^{\frac{1}{3}}}\\ \amp=\frac{1}{5^{2\cdot\frac{1}{3}}}\amp\text{because }(ab)^n=a^nb^n\\ \amp=\frac{1}{5^{\frac{2}{3}}}\\ \amp=5^{-\frac{2}{3}}\amp\text{because }x^{-n}=\frac{1}{x^n} \end{align*}The next few examples simplify expressions with fractional exponents. The key is to convert special numbers into prime factors.
Simplify the following expressions if possible.
\(16^{-\frac{3}{4}}\)
\((-32)^{-\frac{1}{5}}\)
Solution:
\begin{align*} (16)^{-\frac{3}{2}}\amp=(2^4)^{-\frac{3}{2}}\\ \amp=2^{4(-\frac{3}{2})}\\ \amp=2^{-6}\\ \amp=\frac{1}{2^6}\\ \amp=\frac{1}{64} \end{align*}Solution:
\begin{align*} (-32)^{-\frac{1}{5}}\amp=[(-2)^5]^{-\frac{1}{5}}\\ \amp=(-2)^{5(-\frac{1}{5})}\\ \amp=(-2)^{-1}\\ \amp=\frac{1}{-2}\\ \amp=-\frac{1}{2} \end{align*}Sometimes it's easier to simplify radical expressions if we convert radicals to fractional exponents, like in the next example.
Simplify \(\sqrt[3]{64^2}\text{.}\)
It's worth reviewing a property and a common mistake:
\begin{align*} \sqrt{x+y}\amp\ne\sqrt{x}+\sqrt{y}\\ \sqrt{xy}\amp=\sqrt{x}\sqrt{y} \end{align*}The same is also true for radicals of other degrees:
\begin{align*} \sqrt[n]{x+y}\amp\ne\sqrt[n]{x}+\sqrt[n]{y}\\ \sqrt[n]{xy}\amp=\sqrt[n]{x}\sqrt[n]{y} \end{align*}Let's look at the next example.
Convert the following radical expressions into expressions with rational exponents, and simplify them if possible.
\(\sqrt{x^2+4}\)
\(\sqrt{x^2-4}\)
\(\sqrt{(x+4)^2}\text{,}\) where \(x\) is positive
\(\sqrt{4x^2}\text{,}\) where \(x\) is positive
Solution:
\begin{equation*} \sqrt{x^2+4}=(x^2+4)^{\frac{1}{2}} \end{equation*}Solution:
\begin{equation*} \sqrt{x^2-4}=(x^2-4)^{\frac{1}{2}} \end{equation*}Solution:
\begin{equation*} \sqrt{(x+4)^2}=x+4\quad\text{because square and square root cancelled} \end{equation*}Solution:
\begin{equation*} \sqrt{4x^2}=\sqrt{4}\cdot\sqrt{x^2}=2x \end{equation*}Fractional Exponents
With the property \(\sqrt[n]{x}=x^{\frac{1}{n}}\text{,}\) we can simplify radical expressions with exponent properties we learned in Section 6.1. Let's look at a few examples.
Use rational exponent properties to simplify \(\sqrt{y}\cdot\sqrt[4]{y}\text{.}\)
Use rational exponent properties to simplify \(\frac{\sqrt{z}}{\sqrt[3]{z}}\text{.}\)
Use rational exponent properties to simplify \(\sqrt[3]{8p^2}\text{.}\)
Use rational exponent properties to simplify \(\sqrt{\sqrt[4]{q}}\text{.}\)
Use rational exponent properties to simplify \(\sqrt[3]{\frac{r^6}{27}}\text{.}\)
Use rational exponent properties to simplify \((\frac{25}{s^4})^{-\frac{1}{2}}\text{.}\)
Simplifying Expressions with Rational Exponents
