ORCCA Radical Expressions and Rational Exponents

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Section 6.3 Radical Expressions and Rational Exponents

Objectives: PCC Course Content and Outcome Guide

Recall that in Subsection 6.1.3, we learned to evaluate the cube root of a number, say $\sqrt[3]{8}\text{,}$ we can type 8^(1/3) into a calculator. This suggests that $\sqrt[3]{8}=8^{\sfrac{1}{3}}\text{.}$ In this section, we will learn why this is true, and how to simplify expressions with rational exponents.

permalinkMany learners will find a review of exponent rules to be helpful before continuing with the current section. Section 5.2 covers an introduction to exponent rules, and there is more in Section 5.6. The basic rules are summarized in List 5.6.13. These rules are still true and we can use them throughout this section whenever they might help.

Figure 6.3.1. Alternative Video Lesson

Subsection 6.3.1 Radical Expressions and Rational Exponents

permalinkCompare the following calculations:

$\begin{align*} \sqrt{9}\cdot\sqrt{9}\amp=3\cdot3\amp9^{\sfrac{1}{2}}\cdot9^{\sfrac{1}{2}}\amp=9^{\sfrac{1}{2}+\sfrac{1}{2}}\\ \amp=9\amp\amp=9^{1}\\ \amp\amp\amp=9 \end{align*}$

permalinkIf we rewrite the above calculations with exponents, we have:

$\begin{align*} \left(\sqrt{9}\right)^2\amp=9\amp\left(9^{\sfrac{1}{2}}\right)^2\amp=9 \end{align*}$

Since $\sqrt{9}$ and $9^{\sfrac{1}{2}}$ are both positive, and squaring either of them generates the same number, we conclude that:

$\begin{equation*} \sqrt{9}=9^{\sfrac{1}{2}} \end{equation*}$

permalinkWe can verify this result by entering 9^(1/2) into a calculator, and we get 3. In general for any non-negative real number $a\text{,}$ we have:

$\begin{equation*} \sqrt{a}=a^{\sfrac{1}{2}} \end{equation*}$

Similarly, when $a$ is non-negative all of the following are true:

$\begin{align*} \sqrt[2]{a}\amp=a^{\sfrac{1}{2}}\amp\sqrt[3]{a}\amp=a^{\sfrac{1}{3}}\amp\sqrt[4]{a}\amp=a^{\sfrac{1}{4}}\amp\sqrt[5]{a}\amp=a^{\sfrac{1}{5}}\amp\cdots \end{align*}$

For example, when we see $16^{\sfrac{1}{4}}\text{,}$ that is equal to $\sqrt[4]{16}\text{,}$ which we know is $2$ because $\overbrace{2\cdot2\cdot2\cdot2}^{\text{four times}}=16\text{.}$ How can we relate this to the exponential expression $16^{\sfrac{1}{4}}\text{?}$ In a sense, we are cutting up $16$ into $4$ equal parts. But not parts that you add together, rather parts that you multiply together.

permalinkLet's summarize this information with a new exponent rule.

Fact 6.3.2. Radicals and Rational Exponents Rule.

If $m$ is any natural number, and $a$ is any non-negative real number, then

$\begin{equation*} a^{\sfrac{1}{m}}=\sqrt[m]{a}\text{.} \end{equation*}$

Additionally, if $m$ is an odd natural number, then even when $a$ is negative, we still have $a^{\sfrac{1}{m}}=\sqrt[m]{a}\text{.}$

Warning 6.3.3. Exponents on Negative Bases.

Some computers and calculators follow different conventions when there is an exponent on a negative base. To see an example of this, visit WolframAlpha and try entering cuberoot(-8), and then try (-8)^(1/3), and you will get different results. cuberoot(-8) will come out as $-2\text{,}$ but (-8)^(1/3) will come out as a certain non-real complex number. Most likely, any calculator you are using does behave as in Fact 6.3.2, but you should confirm this.

permalinkWith the Radicals and Rational Exponents Rule, we can re-write radical expressions as expressions with rational exponents.

Example 6.3.4.

Write the radical expression $\sqrt[3]{6}$ as an expression with a rational exponent. Then use a calculator to find its decimal approximation.

According to the Radicals and Rational Exponents Rule, $\sqrt[3]{6}=6^{\sfrac{1}{3}}\text{.}$ A calculator tells us that 6^(1/3) works out to approximately $1.817\text{.}$

permalinkFor many examples that follow, we will not need a calculator. We will, however, need to recognize the roots in Figure 6.3.5.

Square Roots	Cube Roots	$4^{\text{th}}$ -Roots	$5^{\text{th}}$ -Roots	Roots of Powers of $2$
$\sqrt{1}=1$	$\sqrt[3]{1}=1$	$\sqrt[4]{1}=1$	$\sqrt[5]{1}=1$
$\sqrt{4}=2$	$\sqrt[3]{8}=2$	$\sqrt[4]{16}=2$	$\sqrt[5]{32}=2$	$\sqrt{4}=2$
$\sqrt{9}=3$	$\sqrt[3]{27}=3$	$\sqrt[4]{81}=3$		$\sqrt[3]{8}=2$
$\sqrt{16}=4$	$\sqrt[3]{64}=4$			$\sqrt[4]{16}=2$
$\sqrt{25}=5$	$\sqrt[3]{125}=5$			$\sqrt[5]{32}=2$
$\sqrt{36}=6$				$\sqrt[6]{64}=2$
$\sqrt{49}=7$				$\sqrt[7]{128}=2$
$\sqrt{64}=8$				$\sqrt[8]{256}=2$
$\sqrt{81}=9$				$\sqrt[9]{512}=2$
$\sqrt{100}=10$				$\sqrt[10]{1024}=2$
$\sqrt{121}=11$
$\sqrt{144}=12$

Figure 6.3.5. Small Roots of Appropriate Natural Numbers

Example 6.3.6.

Write the expressions in radical form using the Radicals and Rational Exponents Rule and simplify the results.

$4^{\sfrac{1}{2}}$
$(-9)^{\sfrac{1}{2}}$
$-16^{\sfrac{1}{4}}$
$64^{-\sfrac{1}{3}}$
$(-27)^{\sfrac{1}{3}}$
$3^{\sfrac{1}{2}}\cdot3^{\sfrac{1}{2}}$

\(\begin{aligned}[t] 4^{\sfrac{1}{2}}\amp=\sqrt{4}\\ \amp=2 \end{aligned}\)
\(\begin{aligned}[t] (-9)^{\sfrac{1}{2}}\amp=\sqrt{-9} \end{aligned}\)This value is non-real.
Without parentheses around \(-16\text{,}\) the negative sign in this problem should be left out of the radical.

\(\begin{aligned}[t] -16^{\sfrac{1}{4}}\amp=-\sqrt[4]{16}\\ \amp=-2 \end{aligned}\)
Here we will use the Negative Exponent Rule.

\(\begin{aligned}[t] 64^{-\sfrac{1}{3}}\amp=\frac{1}{64^{\sfrac{1}{3}}}\\ \amp=\frac{1}{\sqrt[3]{64}}\\ \amp=\frac{1}{4} \end{aligned}\)
\(\begin{aligned}[t] (-27)^{\sfrac{1}{3}}\amp=\sqrt[3]{-27}\\ \amp=-3 \end{aligned}\)
\(\begin{aligned}[t] 3^{\sfrac{1}{2}}\cdot3^{\sfrac{1}{2}}\amp=\sqrt{3}\cdot\sqrt{3}\\ \amp=\sqrt{3\cdot 3}\\ \amp=\sqrt{9}\\ \amp=3 \end{aligned}\)

permalinkThe Radicals and Rational Exponents Rule applies to variables in expressions just as much as it does to numbers.

Example 6.3.7.

Write the expressions as simplified as they can be using radicals.

$2x^{-\sfrac{1}{2}}$
$(5x)^{\sfrac{1}{3}}$
$\left(-27x^{12}\right)^{\sfrac{1}{3}}$
$\left(\frac{16x}{81y^8}\right)^{\sfrac{1}{4}}$

Note that in this example the exponent is only applied to the \(x\text{.}\) Making this type of observation should be our first step for each of these exercises.

\begin{align*} 2x^{-\sfrac{1}{2}}\amp=\frac{2}{x^{\sfrac{1}{2}}} \amp\amp\text{by the }\knowl{./knowl/item-exponent-rules-negative-exponent-definition.html}{\text{Negative Exponent Rule}}\\ \amp=\frac{2}{\sqrt{x}} \amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}} \end{align*}
In this exercise, the exponent applies to both the \(5\) and \(x\text{.}\)

\begin{align*} (5x)^{\sfrac{1}{3}}\amp=\sqrt[3]{5x} \amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}} \end{align*}
We start out as with the previous exercise. As in the previous exercise, we have a choice as to how to simplify this expression. Here we should note that we do know what the cube root of \(-27\) is, so we will take the path to splitting up the expression, using the Product to a Power Rule, before applying the root.

\begin{align*} \left(-27x^{12}\right)^{\sfrac{1}{3}}\amp=\sqrt[3]{-27x^{12}}\\ \end{align*}
Here we notice that \(-27\) has a nice cube root, so it is good to break up the radical.
\begin{align*} \amp=\sqrt[3]{-27}\sqrt[3]{x^{12}}\\ \amp=-3\sqrt[3]{x^{12}} \end{align*}

Can this be simplified more? There are two ways to think about that. One way is to focus on the cube root and see that \(x^4\) cubes to make \(x^{12}\text{,}\) and the other way is to convert the cube root back to a fraction exponent and use exponent rules.

\begin{align*} \amp=-3\sqrt[3]{x^4x^4x^4}\amp\amp=-3\left(x^{12}\right)^{\sfrac{1}{3}}\\ \amp=-3x^4\amp\amp=-3x^{12\cdot\sfrac{1}{3}}\\ \amp\amp\amp=-3x^{4} \end{align*}
We'll use the exponent rule for a fraction raised to a power.

\begin{align*} \left(\frac{16x}{81y^8}\right)^{\sfrac{1}{4}}\amp=\frac{\left(16x\right)^{\sfrac{1}{4}}}{\left(81y^8\right)^{\sfrac{1}{4}}} \amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-quotient-to-a-power.html}{\text{Quotient to a Power Rule}}\\ \amp=\frac{16^{\sfrac{1}{4}}\cdot x^{\sfrac{1}{4}}}{81^{\sfrac{1}{4}} \cdot \left(y^8\right)^{\sfrac{1}{4}}} \amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-product-to-a-power.html}{\text{Product to a Power Rule}}\\ \amp=\frac{16^{\sfrac{1}{4}}\cdot x^{\sfrac{1}{4}}}{81^{\sfrac{1}{4}}\cdot y^2}\\ \amp=\frac{\sqrt[4]{16}\cdot \sqrt[4]{x}}{\sqrt[4]{81}\cdot y^2} \amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=\frac{2\sqrt[4]{x}}{3y^2} \end{align*}

Remark 6.3.8.

In general, it is easier to do algebra with rational exponents on variables than with radicals of variables. You should use Radicals and Rational Exponents Rule to convert from rational exponents to radicals on variables only as a last step in simplifying.

The Radicals and Rational Exponents Rule describes what can be done when there is a fractional exponent and the numerator is a $1\text{.}$ The numerator doesn't have to be a $1$ though and we need guidance for that situation.

Fact 6.3.9. Full Radicals and Rational Exponents Rule.

If $m$ and $n$ are natural numbers such that $\frac{m}{n}$ is a reduced fraction, and $a$ is any non-negative real number, then

$\begin{equation*} a^{\sfrac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m\text{.} \end{equation*}$

Additionally, if $n$ is an odd natural number, then even when $a$ is negative, we still have $a^{\sfrac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m\text{.}$

Example 6.3.10. Guitar Frets.

On a guitar, there are $12$ frets separating a note and the same note one octave higher. By moving from one fret to another that is five frets away, the frequency of the note changes by a factor of $2^{5/12}\text{.}$ Use the Full Radicals and Rational Exponents Rule to write this number as a radical expression. And use a calculator to find this number as a decimal.

According to the Full Radicals and Rational Exponents Rule,

\begin{align*} 2^{5/12}\amp=\sqrt[12]{2^5}\\ \amp=\sqrt[12]{32} \end{align*}

A calculator says \(2^{5/12}\approx1.334\cdots\text{.}\) The fact that this is very close to \(\frac{4}{3}\approx1.333\ldots\) is important. It is part of the explanation for why two notes that are five frets apart on the same string would sound good to human ears when played together as a chord (known as a “fourth,” in music).

Remark 6.3.11.

By the Full Radicals and Rational Exponents Rule, there are two ways to express $a^{\sfrac{m}{n}}$ as a radical expression:

$\begin{align*} a^{\sfrac{m}{n}}\amp=\sqrt[n]{a^m}\amp\amp\text{and}\amp a^{\sfrac{m}{n}}\amp=\left(\sqrt[n]{a}\right)^m \end{align*}$

There are different times to use each formula. In general, use $a^{\sfrac{m}{n}}=\sqrt[n]{a^m}$ for variables and $a^{\sfrac{m}{n}}=\left(\sqrt[n]{a}\right)^m$ for numbers.

Example 6.3.12.

Consider the expression $27^{\sfrac{4}{3}}\text{.}$ Use both versions of the Full Radicals and Rational Exponents Rule to explain why Remark 6.3.11 says that with numbers, $a^{\sfrac{m}{n}}=\left(\sqrt[n]{a}\right)^m$ is preferred.
Consider the expression $x^{\sfrac{4}{3}}\text{.}$ Use both versions of the Full Radicals and Rational Exponents Rule to explain why Remark 6.3.11 says that with variables, $a^{\sfrac{m}{n}}=\sqrt[n]{a^m}$ is preferred.

The expression \(27^{\sfrac{4}{3}}\) can be evaluated in the following two ways.

\begin{align*} 27^{\sfrac{4}{3}}\amp=\sqrt[3]{27^4}\amp\amp\text{by the first part of the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Full Radicals and Rational Exponents Rule}}\\ \amp=\sqrt[3]{531441}\\ \amp=81\\ \amp\amp\text{or}\\ 27^{\sfrac{4}{3}}\amp=\left(\sqrt[3]{27}\right)^4\amp\amp\text{by the second part of the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Full Radicals and Rational Exponents Rule}}\\ \amp=3^4\\ \amp=81 \end{align*}

The calculation using \(a^{\sfrac{m}{n}}=\left(\sqrt[n]{a}\right)^m\) worked with smaller numbers and can be done without a calculator. This is why we made the general recommendation in Remark 6.3.11.
The expression \(x^{\sfrac{4}{3}}\) can be evaluated in the following two ways.

\begin{align*} x^{\sfrac{4}{3}}\amp=\sqrt[3]{x^4}\amp\amp\text{by the first part of } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Full Radicals and Rational Exponents Rule}}\\ \amp\amp\text{or}\\ x^{\sfrac{4}{3}}\amp=\left(\sqrt[3]{x}\right)^4\amp\amp\text{by the second part of the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Full Radicals and Rational Exponents Rule}} \end{align*}

In this case, the simplification using \(a^{\sfrac{m}{n}}=\sqrt[n]{a^m}\) is just shorter looking and easier to write. This is why we made the general recommendation in Remark 6.3.11.

Example 6.3.13.

Simplify the expressions using Fact 6.3.9.

$8^{\sfrac{2}{3}}$
$(64x)^{-\sfrac{2}{3}}$
$\left(-\frac{27}{64}\right)^{\sfrac{2}{3}}$

We will use the second part of the Full Radicals and Rational Exponents Rule, since this expression only involves a number base (not variable).

\(\begin{aligned} 8^{\sfrac{2}{3}}\amp=\left(\sqrt[3]{8}\right)^2\\ \amp=2^2\\ \amp=4 \end{aligned}\)
\(\begin{aligned}[t] (64x)^{-\sfrac{2}{3}}\amp=\frac{1}{(64x)^{\sfrac{2}{3}}}\\ \amp=\frac{1}{64^{\sfrac{2}{3}}x^{\sfrac{2}{3}}}\\ \amp=\frac{1}{\left(\sqrt[3]{64}\right)^2\sqrt[3]{x^2}}\\ \amp=\frac{1}{4^2\sqrt[3]{x^2}}\\ \amp=\frac{1}{16\sqrt[3]{x^2}} \end{aligned}\)
In this problem the negative can be associated with either the numerator or the denominator, but not both. We choose the numerator.

\begin{align*} \left(-\frac{27}{64}\right)^{\sfrac{2}{3}}\amp=\left(\sqrt[3]{-\frac{27}{64}}\right)^2 \amp\amp\text{by the second part of the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Full Radicals and Rational Exponents Rule}}\\ \amp=\left(\frac{\sqrt[3]{-27}}{\sqrt[3]{64}}\right)^2\\ \amp=\left(\frac{-3}{4}\right)^2\\ \amp=\frac{(-3)^2}{(4)^2}\\ \amp=\frac{9}{16} \end{align*}

Subsection 6.3.2 More Expressions with Rational Exponents

permalinkTo recap, here is a “complete” list of exponent and radical rules.

List 6.3.14. Complete List of Exponent Rules

Product Rule: $a^{n} \cdot a^{m} = a^{n+m}$
Power to a Power Rule: $(a^{n})^{m} = a^{n\cdot m}$
Product to a Power Rule: $(ab)^{n} = a^{n} \cdot b^{n}$
Quotient Rule: $\dfrac{a^{n}}{a^{m}} = a^{n-m}\text{,}$ as long as $a \neq 0$
Quotient to a Power Rule: $\left( \dfrac{a}{b} \right)^{n} = \dfrac{a^{n}}{b^{n}}\text{,}$ as long as $b \neq 0$
Zero Exponent Rule: $a^{0} = 1$ for $a\neq0$
Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$
Negative Exponent Reciprocal Rule: $\frac{1}{a^{-n}} = a^n$
Negative Exponent on Fraction Rule: $\left(\frac{x}{y}\right)^{-n}=\left(\frac{y}{x}\right)^{n}$
Radical and Rational Exponent Rule: $x^{\sfrac{1}{n}}=\sqrt[n]{x}$
Radical and Rational Exponent Rule: $x^{\sfrac{m}{n}}=\left(\sqrt[n]{x}\right)^m\text{,}$ usually for numbers
Radical and Rational Exponent Rule: $x^{\sfrac{m}{n}}=\sqrt[n]{x^m}\text{,}$ usually for variables

Example 6.3.15.

Convert the following radical expressions into expressions with rational exponents, and simplify them if possible.

$\dfrac{1}{\sqrt{x}}$
$\dfrac{1}{\sqrt[3]{25}}$

\begin{align*} \frac{1}{\sqrt{x}}\amp=\frac{1}{x^{\sfrac{1}{2}}}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=x^{-\sfrac{1}{2}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-negative-exponent-definition.html}{\text{Negative Exponent Rule}} \end{align*}
\begin{align*} \frac{1}{\sqrt[3]{25}}\amp=\frac{1}{25^{\sfrac{1}{3}}}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=\frac{1}{\left(5^2\right)^{\sfrac{1}{3}}}\\ \amp=\frac{1}{5^{2\cdot\sfrac{1}{3}}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-power-to-a-power.html}{\text{Power to a Power Rule}}\\ \amp=\frac{1}{5^{\sfrac{2}{3}}}\\ \amp=5^{-\sfrac{2}{3}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-negative-exponent-definition.html}{\text{Negative Exponent Rule}} \end{align*}

permalinkLearners of these simplifications often find it challenging, so we now include a many examples of varying difficulty.

Example 6.3.16.

Use exponent properties in List 6.3.14 to simplify the expressions, and write all final versions using radicals.

$2w^{\sfrac{7}{8}}$
$\frac{1}{2}y^{-\sfrac{1}{2}}$
$\left(27b\right)^{\sfrac{2}{3}}$
$\left(-8p^6\right)^{\sfrac{5}{3}}$
$\sqrt{x^3}\cdot\sqrt[4]{x}$
$h^{\sfrac{1}{3}}+h^{\sfrac{1}{3}}+h^{\sfrac{1}{3}}$
$\frac{\sqrt{z}}{\sqrt[3]{z}}$
$\sqrt{\sqrt[4]{q}}$
$3\left(c^{\sfrac{1}{2}}+d^{\sfrac{1}{2}}\right)^2$
$3\left(4k^{\sfrac{2}{3}}\right)^{-\sfrac{1}{2}}$

\begin{align*} 2w^{\sfrac{7}{8}}\amp=2\sqrt[8]{w^7}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Full Radicals and Rational Exponents Rule}} \end{align*}
\begin{align*} \frac{1}{2}y^{-\sfrac{1}{2}}\amp=\frac{1}{2}\frac{1}{y^{\sfrac{1}{2}}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-negative-exponent-definition.html}{\text{Negative Exponent Rule}}\\ \amp=\frac{1}{2}\frac{1}{\sqrt{y}}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Full Radicals and Rational Exponents Rule}}\\ \amp=\frac{1}{2\sqrt{y}} \end{align*}
\begin{align*} \left(27b\right)^{\sfrac{2}{3}}\amp=\left(27\right)^{\sfrac{2}{3}}\cdot\left(b\right)^{\sfrac{2}{3}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-product-to-a-power.html}{\text{Product to a Power Rule}}\\ \amp=\left(\sqrt[3]{27}\right)^2\cdot\sqrt[3]{b^2}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Full Radicals and Rational Exponents Rule}}\\ \amp=3^2\cdot\sqrt[3]{b^2}\\ \amp=9\sqrt[3]{b^2} \end{align*}
\begin{align*} \left(-8p^6\right)^{\sfrac{5}{3}}\amp=\left(-8\right)^{\sfrac{5}{3}}\cdot\left(p^6\right)^{\sfrac{5}{3}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-product-to-a-power.html}{\text{Product to a Power Rule}}\\ \amp=\left(-8\right)^{\sfrac{5}{3}}\cdot p^{6\cdot\sfrac{5}{3}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-power-to-a-power.html}{\text{Power to a Power Rule}}\\ \amp=\left(\sqrt[3]{-8}\right)^5\cdot p^{10}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Full Radicals and Rational Exponents Rule}}\\ \amp=(-2)^5\cdot p^{10}\\ \amp=-32p^{10} \end{align*}
\begin{align*} \sqrt{x^3}\cdot\sqrt[4]{x}\amp=x^{\sfrac{3}{2}}\cdot x^{\sfrac{1}{4}}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Full Radicals and Rational Exponents Rule}}\\ \amp=x^{\sfrac{3}{2}+\sfrac{1}{4}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-product.html}{\text{Product Rule}}\\ \amp=x^{\sfrac{6}{4}+\sfrac{1}{4}}\\ \amp=x^{\sfrac{7}{4}}\\ \amp=\sqrt[4]{x^7}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Full Radicals and Rational Exponents Rule}} \end{align*}
\begin{align*} h^{\sfrac{1}{3}}+h^{\sfrac{1}{3}}+h^{\sfrac{1}{3}}\amp=3h^{\sfrac{1}{3}}\\ \amp=3\sqrt[3]{h}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}} \end{align*}
\begin{align*} \frac{\sqrt{z}}{\sqrt[3]{z}}\amp=\frac{z^{\sfrac{1}{2}}}{z^{\sfrac{1}{3}}}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=z^{\sfrac{1}{2}-\sfrac{1}{3}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-quotient.html}{\text{Quotient Rule}}\\ \amp=z^{\sfrac{3}{6}-\sfrac{2}{6}}\\ \amp=z^{\sfrac{1}{6}}\\ \amp=\sqrt[6]{z}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}} \end{align*}
\begin{align*} \sqrt{\sqrt[4]{q}}\amp=\sqrt{q^{\sfrac{1}{4}}}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=\left(q^{\sfrac{1}{4}}\right)^{\sfrac{1}{2}}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=q^{\sfrac{1}{4}\cdot\sfrac{1}{2}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-power-to-a-power.html}{\text{Power to a Power Rule}}\\ \amp=q^{\sfrac{1}{8}}\\ \amp=\sqrt[8]{q}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}} \end{align*}
\begin{alignat*}{2} 3\left(c^{\sfrac{1}{2}}+d^{\sfrac{1}{2}}\right)^2\amp=3\left(c^{\sfrac{1}{2}}+d^{\sfrac{1}{2}}\right)\left(c^{\sfrac{1}{2}}+d^{\sfrac{1}{2}}\right)\\ \amp=3\left(\left(c^{\sfrac{1}{2}}\right)^2+2c^{\sfrac{1}{2}}\cdot d^{\sfrac{1}{2}}+\left(d^{\sfrac{1}{2}}\right)^2\right)\\ \amp=3\left(c^{\sfrac{1}{2}\cdot 2}+2c^{\sfrac{1}{2}}\cdot d^{\sfrac{1}{2}}+d^{\sfrac{1}{2}\cdot 2}\right)\\ \amp=3\left(c+2c^{\sfrac{1}{2}}\cdot d^{\sfrac{1}{2}}+d\right)\\ \amp=3\left(c+2(cd)^{\sfrac{1}{2}}+d\right)\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-product-to-a-power.html}{\text{Product to a Power Rule}}\\ \amp=3\left(c+2\sqrt{cd}+d\right)\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=3c+6\sqrt{cd}+3d \end{alignat*}
\begin{align*} 3\left(4k^{\sfrac{2}{3}}\right)^{-\sfrac{1}{2}}\amp=\frac{3}{\left(4k^{\sfrac{2}{3}}\right)^{\sfrac{1}{2}}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-negative-exponent-definition.html}{\text{Negative Exponent Rule}}\\ \amp=\frac{3}{4^{\sfrac{1}{2}}\left(k^{\sfrac{2}{3}}\right)^{\sfrac{1}{2}}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-product-to-a-power.html}{\text{Product to a Power Rule}}\\ \amp=\frac{3}{4^{\sfrac{1}{2}}k^{\sfrac{2}{3}\cdot\sfrac{1}{2}}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-power-to-a-power.html}{\text{Power to a Power Rule}}\\ \amp=\frac{3}{4^{\sfrac{1}{2}}k^{\sfrac{1}{3}}}\\ \amp=\frac{3}{\sqrt{4}\cdot\sqrt[3]{k}}\amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=\frac{3}{2\sqrt[3]{k}} \end{align*}

permalinkWe will end a with a short application of rational exponents. Kepler's Laws of Orbital Motion ¹en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motion describe how planets orbit stars and how satellites orbit planets. In particular, his third law has a rational exponent, which we will now explore.

Example 6.3.17. Kepler and the Satellite.

Kepler's third law of motion says that for objects with a roughly circular orbit that the time (in hours) that it takes to make one full revolution around the planet, $T\text{,}$ is proportional to three-halves power of the distance (in kilometers) from the center of the planet to the satellite, $r\text{.}$ For the Earth, it looks like this:

$\begin{equation*} T=\frac{2\pi}{\sqrt{G\cdot M_E}}r^{\sfrac{3}{2}} \end{equation*}$

In this case, both $G$ and $M_E$ are constants. $G$ stands for the universal gravitational constant ² where $G$ is about $8.65\times 10^{-13}$ and $M_E$ stands for the mass of the Earth ³ where $M_E$ is about $5.972\times 10^{24}$ kg. Inputting these values into this formula yields a simplified version that looks like this:

$\begin{equation*} T\approx2.76\times 10^{-6}r^{\sfrac{3}{2}} \end{equation*}$

Most satellites orbit in what is called low Earth orbit ⁴en.wikipedia.org/wiki/Low_Earth_orbit, including the international space station which orbits at about 340 km above from Earth's surface. The Earth's average radius is about 6380 km. Find the period of the international space station.

The formula has already been identified, but the input takes just a little thought. The formula uses \(r\) as the distance from the center of the Earth to the satellite, so to find \(r\) we need to combine the radius of the Earth and the distance to the satellite above the surface of the Earth.

\begin{align*} r\amp=340+6380\\ \amp=6720 \end{align*}

Now we can input this value into the formula and evaluate.

\begin{align*} T\amp\approx2.76\cdot 10^{-6}r^{\sfrac{3}{2}}\\ \amp\approx2.76\cdot 10^{-6}(\substitute{6720})^{\sfrac{3}{2}}\\ \amp\approx2.76\cdot 10^{-6}\left(\sqrt{6720}\right)^3\\ \amp\approx1.52 \end{align*}

The formula tells us that it takes a little more than an hour and a half for the ISS to orbit the Earth! That works out to 15 or 16 sunrises per day.

Reading Questions 6.3.3 Reading Questions

1.

Raising a number to a reciprocal power (like $\frac{1}{2}$ or $\frac{1}{5}$ ) is the same as doing what other thing to that number?

2.

When the exponent on an expression is a fraction like $\frac{3}{5}\text{,}$ which part of the fraction is essentially the index of a radical?

Exercises 6.3.4 Exercises

Review and Warmup

1.

Use the properties of exponents to simplify the expression.

${x^{13}}\cdot{x^{17}}$

2.

Use the properties of exponents to simplify the expression.

${y^{15}}\cdot{y^{11}}$

3.

Use the properties of exponents to simplify the expression.

$\displaystyle{\left(t^{11}\right)^{3}}$

4.

Use the properties of exponents to simplify the expression.

$\displaystyle{\left(y^{12}\right)^{10}}$

5.

Use the properties of exponents to simplify the expression.

$\left(\displaystyle\frac{7x^{2}}{2}\right)^{2}=$

6.

Use the properties of exponents to simplify the expression.

$\left(\displaystyle\frac{3x^{3}}{8}\right)^{2}=$

7.

Use the properties of exponents to simplify the expression.

$\left(-6y^{4}\right)^3$

8.

Use the properties of exponents to simplify the expression.

$\left(-2x^{6}\right)^2$

9.

Use the properties of exponents to simplify the expression.

$\displaystyle\frac{{y^{11}}}{{y}}=$

10.

Use the properties of exponents to simplify the expression.

$\displaystyle\frac{{t^{13}}}{{t^{9}}}=$

11.

Rewrite the expression simplified and using only positive exponents.

$r^{-9}\cdot r^{3}=$

12.

Rewrite the expression simplified and using only positive exponents.

$t^{-3}\cdot t^{2}=$

13.

Rewrite the expression simplified and using only positive exponents.

$(9t^{-14})\cdot (10t^{2})=$

14.

Rewrite the expression simplified and using only positive exponents.

$(6x^{-8})\cdot (5x^{4})=$

Calculations

15.

Without using a calculator, evaluate the expression.

$\displaystyle{ 9^{\frac{1}{2}}= }$
$\displaystyle{ (-9)^{\frac{1}{2}}= }$
$\displaystyle{ -9^{\frac{1}{2}}= }$

16.

Without using a calculator, evaluate the expression.

$\displaystyle{ 16^{\frac{1}{2}}= }$
$\displaystyle{ (-16)^{\frac{1}{2}}= }$
$\displaystyle{ -16^{\frac{1}{2}}= }$

17.

Without using a calculator, evaluate the expression.

$\displaystyle{ 8^{\frac{1}{3}}= }$
$\displaystyle{ (-8)^{\frac{1}{3}}= }$
$\displaystyle{ -8^{\frac{1}{3}}= }$

18.

Without using a calculator, evaluate the expression.

$\displaystyle{ 27^{\frac{1}{3}}= }$
$\displaystyle{ (-27)^{\frac{1}{3}}= }$
$\displaystyle{ -27^{\frac{1}{3}}= }$

19.

Without using a calculator, evaluate the expression.

$\displaystyle{ 9^{-\frac{3}{2}} = }$

20.

Without using a calculator, evaluate the expression.

$\displaystyle{ 125^{-\frac{1}{3}} = }$

21.

Without using a calculator, evaluate the expression.

$\displaystyle{ \left(\frac{1}{81}\right)^{-\frac{3}{4}} = }$

22.

Without using a calculator, evaluate the expression.

$\displaystyle{ \left(\frac{1}{9}\right)^{-\frac{3}{2}} = }$

23.

Without using a calculator, evaluate the expression.

$\displaystyle{ \sqrt[2]{9^{3}}= }$

24.

Without using a calculator, evaluate the expression.

$\displaystyle{ \sqrt[2]{81^{3}}= }$

25.

Without using a calculator, evaluate the expression.

$\displaystyle{ \sqrt[5]{1024}= }$

26.

Without using a calculator, evaluate the expression.

$\displaystyle{ \sqrt[3]{64}= }$

27.

Without using a calculator, evaluate the expression.

$\displaystyle{\sqrt[3]{8} =}$
$\displaystyle{\sqrt[3]{-8} =}$
$\displaystyle{-\sqrt[3]{8} =}$

28.

Without using a calculator, evaluate the expression.

$\displaystyle{\sqrt[3]{27} =}$
$\displaystyle{\sqrt[3]{-27} =}$
$\displaystyle{-\sqrt[3]{27} =}$

29.

Without using a calculator, evaluate the expression.

$\displaystyle{\sqrt[4]{16} =}$
$\displaystyle{\sqrt[4]{-16} =}$
$\displaystyle{-\sqrt[4]{16} =}$

30.

Without using a calculator, evaluate the expression.

$\displaystyle{\sqrt[4]{81} =}$
$\displaystyle{\sqrt[4]{-81} =}$
$\displaystyle{-\sqrt[4]{81} =}$

31.

Without using a calculator, evaluate the expression.

$\displaystyle{ \sqrt[3]{{-{\frac{27}{125}}}}= }$ .

32.

Without using a calculator, evaluate the expression.

$\displaystyle{ \sqrt[3]{{-{\frac{27}{125}}}}= }$ .

33.

Without using a calculator, evaluate the expression.

$\displaystyle{ \sqrt[3]{{-{\frac{1}{64}}}}= }$ .

34.

Without using a calculator, evaluate the expression.

$\displaystyle{ \sqrt[3]{{-{\frac{27}{125}}}}= }$ .

35.

Use a calculator to evaluate the expression as a decimal to four significant digits.

$\displaystyle{\sqrt[3]{9^{2}}=}$

36.

Use a calculator to evaluate the expression as a decimal to four significant digits.

$\displaystyle{\sqrt[5]{11^{3}}=}$

37.

Use a calculator to evaluate the expression as a decimal to four significant digits.

$\displaystyle{\sqrt[3]{13^{2}}=}$

38.

Use a calculator to evaluate the expression as a decimal to four significant digits.

$\displaystyle{\sqrt[5]{18^{3}}=}$

39.

On a guitar, there are $12$ frets separating a note and the same note one octave higher. By moving from one fret to another that is seven frets away, the frequency of the note changes by a factor of $2^{7/12}\text{.}$ Use a calculator to find this number as a decimal.

This decimal shows you that $2^{7/12}$ is very close to a “nice” fraction with small numerator and denominator. Notes with this frequency ratio form a “perfect fifth” in music. What is that fraction?

40.

On a guitar, there are $12$ frets separating a note and the same note one octave higher. By moving from one fret to another that is four frets away, the frequency of the note changes by a factor of $2^{4/12}\text{.}$ Use a calculator to find this number as a decimal.

This decimal shows you that $2^{4/12}$ is very close to a “nice” fraction with small numerator and denominator. Notes with this frequency ratio form a “major third” in music. What is that fraction?

Convert Radicals to Fractional Exponents

41.

Use rational exponents to write the expression.

$\displaystyle{\sqrt[9]{x}}$ =

42.

Use rational exponents to write the expression.

$\displaystyle{\sqrt[6]{y}}$ =

43.

Use rational exponents to write the expression.

$\displaystyle{\sqrt[3]{4 z + 6}=}$

44.

Use rational exponents to write the expression.

$\displaystyle{\sqrt{9 t + 10}=}$

45.

Use rational exponents to write the expression.

$\displaystyle{ \sqrt[6]{r} = }$

46.

Use rational exponents to write the expression.

$\displaystyle{ \sqrt[3]{m} = }$

47.

Use rational exponents to write the expression.

$\displaystyle{ \frac{1}{\sqrt[8]{n^{3}}} = }$

48.

Use rational exponents to write the expression.

$\displaystyle{ \frac{1}{\sqrt[5]{b^{4}}} = }$

Convert Fractional Exponents to Radicals

49.

Convert the expression to radical notation.

$\displaystyle{{c^{\frac{2}{3}}}}$ =

50.

Convert the expression to radical notation.

$\displaystyle{{x^{\frac{5}{6}}}}$ =

51.

Convert the expression to radical notation.

$\displaystyle{{y^{\frac{5}{9}}}}$ =

52.

Convert the expression to radical notation.

$\displaystyle{{r^{\frac{2}{3}}}}$ =

53.

Convert the expression to radical notation.

$\displaystyle{{15^{\frac{1}{6}}t^{\frac{5}{6}}}}$ =

54.

Convert the expression to radical notation.

$\displaystyle{{4^{\frac{1}{4}}r^{\frac{3}{4}}}}$ =

55.

Convert $m^{\frac{2}{3}}$ to a radical expression.

56.

Convert $n^{\frac{5}{6}}$ to a radical expression.

57.

Convert $b^{-\frac{3}{5}}$ to a radical expression.

58.

Convert $c^{-\frac{2}{7}}$ to a radical expression.

59.

Convert $2^{\frac{1}{5}} x^{\frac{4}{5}}$ to a radical expression.

60.

Convert $7^{\frac{1}{7}} y^{\frac{3}{7}}$ to a radical expression.

Simplifying Expressions with Rational Exponents

61.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\sqrt[11]{z}\,\sqrt[11]{z}=}$

62.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\sqrt[9]{t}\,\sqrt[9]{t}=}$

63.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\sqrt[5]{32 r^{2}}=}$

64.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\sqrt[3]{27 m^{5}}=}$

65.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\frac{\sqrt[3]{27 n}}{\sqrt[6]{n^{5}}}=}$

66.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\frac{\sqrt{36 b}}{\sqrt[6]{b^{5}}}=}$

67.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\frac{\sqrt{4 c^{3}}}{\sqrt[10]{c}}=}$

68.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\frac{\sqrt{49 x}}{\sqrt[6]{x^{5}}}=}$

69.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\sqrt[5]{y}\cdot\sqrt[10]{y^{3}}=}$

70.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\sqrt{z} \cdot \sqrt[6]{z^{5}}=}$

71.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\sqrt{\sqrt[3]{t}}=}$

72.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\sqrt[4]{\sqrt{r}}=}$

73.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\sqrt{b}\sqrt[7]{b}=}$

74.

Simplify the expression, answering with rational exponents and not radicals.

$\displaystyle{\sqrt{r}\sqrt[8]{r}=}$