Section 6.3 Radical Expressions and Rational Exponents
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.Many learners will find a review of exponent properties to be helpful before continuing with the current section.
[cross-reference to target(s) "section-introduction-to-exponent-properties" missing or not unique] covers an introduction to exponent properties, and there is more in Section 5.6. The basic properties are summarized in List 5.6.13. These properties are still true and we can use them throughout this section whenever they might help.Subsection 6.3.1 Radical Expressions and Rational Exponents
Compare 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*}
If 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*}
We 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.
Let’s summarize this information with a new exponent property.
Fact 6.3.2. Radicals and Rational Exponents Property.
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
1
www.wolframalpha.comcuberoot(-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 2, but you should confirm this.With the Radicals and Rational Exponents Property, 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 Property, \(\sqrt[3]{6}=6^{\sfrac{1}{3}}\text{.}\) A calculator tells us that
6^(1/3) works out to approximately \(1.817\text{.}\)
For many examples that follow, we will not need a calculator. We will, however, need to recognize the roots in Figure 5.
| Square Roots | Cube Roots | \(4^{\text{th}}\)-Roots | \(5^{\text{th}}\)-Roots | Roots of Powers of \(2\) |
| \(\sqrt{1}=1\) | \(\sqrt[3]{1}=1\) | \(\sqrt[4]{1}=1\) | \(\sqrt[5]{1}=1\) | |
| \(\sqrt{4}=2\) | \(\sqrt[3]{8}=2\) | \(\sqrt[4]{16}=2\) | \(\sqrt[5]{32}=2\) | \(\sqrt{4}=2\) |
| \(\sqrt{9}=3\) | \(\sqrt[3]{27}=3\) | \(\sqrt[4]{81}=3\) | \(\sqrt[3]{8}=2\) | |
| \(\sqrt{16}=4\) | \(\sqrt[3]{64}=4\) | \(\sqrt[4]{16}=2\) | ||
| \(\sqrt{25}=5\) | \(\sqrt[3]{125}=5\) | \(\sqrt[5]{32}=2\) | ||
| \(\sqrt{36}=6\) | \(\sqrt[6]{64}=2\) | |||
| \(\sqrt{49}=7\) | \(\sqrt[7]{128}=2\) | |||
| \(\sqrt{64}=8\) | \(\sqrt[8]{256}=2\) | |||
| \(\sqrt{81}=9\) | \(\sqrt[9]{512}=2\) | |||
| \(\sqrt{100}=10\) | \(\sqrt[10]{1024}=2\) | |||
| \(\sqrt{121}=11\) | ||||
| \(\sqrt{144}=12\) |
Example 6.3.6.
Write the expressions in radical form using the Radicals and Rational Exponents Property and simplify the results.
- \(\displaystyle 4^{\sfrac{1}{2}}\)
- \(\displaystyle (-9)^{\sfrac{1}{2}}\)
- \(\displaystyle -16^{\sfrac{1}{4}}\)
- \(\displaystyle 64^{-\sfrac{1}{3}}\)
- \(\displaystyle (-27)^{\sfrac{1}{3}}\)
- \(\displaystyle 3^{\sfrac{1}{2}}\cdot3^{\sfrac{1}{2}}\)
Explanation.
- \(\displaystyle \begin{aligned}[t] 4^{\sfrac{1}{2}}\amp=\sqrt{4}\\ \amp=2 \end{aligned}\)
- \(\begin{aligned}[t] (-9)^{\sfrac{1}{2}}\amp=\sqrt{-9} \end{aligned}\)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 Definition.\(\begin{aligned}[t] 64^{-\sfrac{1}{3}}\amp=\frac{1}{64^{\sfrac{1}{3}}}\\ \amp=\frac{1}{\sqrt[3]{64}}\\ \amp=\frac{1}{4} \end{aligned}\)
- \(\displaystyle \begin{aligned}[t] (-27)^{\sfrac{1}{3}}\amp=\sqrt[3]{-27}\\ \amp=-3 \end{aligned}\)
- \(\displaystyle \begin{aligned}[t] 3^{\sfrac{1}{2}}\cdot3^{\sfrac{1}{2}}\amp=\sqrt{3}\cdot\sqrt{3}\\ \amp=\sqrt{3\cdot 3}\\ \amp=\sqrt{9}\\ \amp=3 \end{aligned}\)
The Radicals and Rational Exponents Property 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.
- \(\displaystyle 2x^{-\sfrac{1}{2}}\)
- \(\displaystyle (5x)^{\sfrac{1}{3}}\)
- \(\displaystyle \left(-27x^{12}\right)^{\sfrac{1}{3}}\)
- \(\displaystyle \left(\frac{16x}{81y^8}\right)^{\sfrac{1}{4}}\)
Explanation.
- 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/xref/item-exponent-properties-negative-exponent-definition.html}{\text{Negative Exponent Definition}}\\ \amp=\frac{2}{\sqrt{x}} \amp\amp\text{by the } \knowl{./knowl/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}} \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/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}} \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 properties.\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 property 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/xref/item-exponent-properties-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/xref/item-exponent-properties-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/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \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 Property to convert from rational exponents to radicals on variables only as a last step in simplifying.
The Radicals and Rational Exponents Property 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 Property.
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 Property to write this number as a radical expression. And use a calculator to find this number as a decimal.
Explanation.
According to the Full Radicals and Rational Exponents Property,
\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 Property, 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 Property to explain why Remark 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 Property to explain why Remark 11 says that with variables, \(a^{\sfrac{m}{n}}=\sqrt[n]{a^m}\) is preferred.
Explanation.
- 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/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}}\\ \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/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}}\\ \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 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/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}}\\ \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/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}} \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 11.
Example 6.3.13.
Simplify the expressions using Fact 9.
- \(\displaystyle 8^{\sfrac{2}{3}}\)
- \(\displaystyle (64x)^{-\sfrac{2}{3}}\)
- \(\displaystyle \left(-\frac{27}{64}\right)^{\sfrac{2}{3}}\)
Explanation.
-
We will use the second part of the Full Radicals and Rational Exponents Property, since this expression only involves a number base (not variable).\(\begin{aligned} 8^{\sfrac{2}{3}}\amp=\left(\sqrt[3]{8}\right)^2\\ \amp=2^2\\ \amp=4 \end{aligned}\)
- \(\displaystyle \begin{aligned}[t] (64x)^{-\sfrac{2}{3}}\amp=\frac{1}{(64x)^{\sfrac{2}{3}}}\\ \amp=\frac{1}{64^{\sfrac{2}{3}}x^{\sfrac{2}{3}}}\\ \amp=\frac{1}{\left(\sqrt[3]{64}\right)^2\sqrt[3]{x^2}}\\ \amp=\frac{1}{4^2\sqrt[3]{x^2}}\\ \amp=\frac{1}{16\sqrt[3]{x^2}} \end{aligned}\)
-
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/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}}\\ \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
To recap, here is a “complete” list of exponent and radical properties.
- Product Property
- \(\displaystyle a^{n} \cdot a^{m} = a^{n+m}\)
- Power to a Power Property
- \(\displaystyle (a^{n})^{m} = a^{n\cdot m}\)
- Product to a Power Property
- \(\displaystyle (ab)^{n} = a^{n} \cdot b^{n}\)
- Quotient Property
- \(\dfrac{a^{n}}{a^{m}} = a^{n-m}\text{,}\) as long as \(a \neq 0\)
- Quotient to a Power Property
- \(\left( \dfrac{a}{b} \right)^{n} = \dfrac{a^{n}}{b^{n}}\text{,}\) as long as \(b \neq 0\)
- Zero Exponent Property
- \(a^{0} = 1\) for \(a\neq0\)
- Negative Exponent Property
- \(\displaystyle a^{-n} = \frac{1}{a^n}\)
- Negative Exponent Reciprocal Property
- \(\displaystyle \frac{1}{a^{-n}} = a^n\)
- Negative Exponent on Fraction Property
- \(\displaystyle \left(\frac{x}{y}\right)^{-n}=\left(\frac{y}{x}\right)^{n}\)
- Radical and Rational Exponent Property
- \(\displaystyle x^{\sfrac{1}{n}}=\sqrt[n]{x}\)
- Radical and Rational Exponent Property
- \(x^{\sfrac{m}{n}}=\left(\sqrt[n]{x}\right)^m\text{,}\) usually for numbers
- Radical and Rational Exponent Property
- \(x^{\sfrac{m}{n}}=\sqrt[n]{x^m}\text{,}\) usually for variables
Example 6.3.15.
Convert the following radical expressions into expressions with rational exponents, and simplify them if possible.
- \(\displaystyle \dfrac{1}{\sqrt{x}}\)
- \(\displaystyle \dfrac{1}{\sqrt[3]{25}}\)
Explanation.
- \begin{align*} \frac{1}{\sqrt{x}}\amp=\frac{1}{x^{\sfrac{1}{2}}}\amp\amp\text{by the } \knowl{./knowl/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \amp=x^{-\sfrac{1}{2}}\amp\amp\text{by the } \knowl{./knowl/xref/item-exponent-properties-negative-exponent-definition.html}{\text{Negative Exponent Definition}} \end{align*}
- \begin{align*} \frac{1}{\sqrt[3]{25}}\amp=\frac{1}{25^{\sfrac{1}{3}}}\amp\amp\text{by the } \knowl{./knowl/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \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/xref/item-exponent-properties-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/xref/item-exponent-properties-negative-exponent-definition.html}{\text{Negative Exponent Definition}} \end{align*}
Learners 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 14 to simplify the expressions, and write all final versions using radicals.
- \(\displaystyle 2w^{\sfrac{7}{8}}\)
- \(\displaystyle \frac{1}{2}y^{-\sfrac{1}{2}}\)
- \(\displaystyle \left(27b\right)^{\sfrac{2}{3}}\)
- \(\displaystyle \left(-8p^6\right)^{\sfrac{5}{3}}\)
- \(\displaystyle \sqrt{x^3}\cdot\sqrt[4]{x}\)
- \(\displaystyle h^{\sfrac{1}{3}}+h^{\sfrac{1}{3}}+h^{\sfrac{1}{3}}\)
- \(\displaystyle \frac{\sqrt{z}}{\sqrt[3]{z}}\)
- \(\displaystyle \sqrt{\sqrt[4]{q}}\)
- \(\displaystyle 3\left(c^{\sfrac{1}{2}}+d^{\sfrac{1}{2}}\right)^2\)
- \(\displaystyle 3\left(4k^{\sfrac{2}{3}}\right)^{-\sfrac{1}{2}}\)
Explanation.
- \begin{align*} 2w^{\sfrac{7}{8}}\amp=2\sqrt[8]{w^7}\amp\amp\text{by the } \knowl{./knowl/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}} \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/xref/item-exponent-properties-negative-exponent-definition.html}{\text{Negative Exponent Definition}}\\ \amp=\frac{1}{2}\frac{1}{\sqrt{y}}\amp\amp\text{by the } \knowl{./knowl/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}}\\ \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/xref/item-exponent-properties-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/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}}\\ \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/xref/item-exponent-properties-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/xref/item-exponent-properties-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/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}}\\ \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/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}}\\ \amp=x^{\sfrac{3}{2}+\sfrac{1}{4}}\amp\amp\text{by the } \knowl{./knowl/xref/item-exponent-properties-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/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}} \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/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}} \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/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \amp=z^{\sfrac{1}{2}-\sfrac{1}{3}}\amp\amp\text{by the } \knowl{./knowl/xref/item-exponent-properties-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/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}} \end{align*}
- \begin{align*} \sqrt{\sqrt[4]{q}}\amp=\sqrt{q^{\sfrac{1}{4}}}\amp\amp\text{by the } \knowl{./knowl/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \amp=\left(q^{\sfrac{1}{4}}\right)^{\sfrac{1}{2}}\amp\amp\text{by the } \knowl{./knowl/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \amp=q^{\sfrac{1}{4}\cdot\sfrac{1}{2}}\amp\amp\text{by the } \knowl{./knowl/xref/item-exponent-properties-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/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}} \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/xref/item-exponent-properties-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/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \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/xref/item-exponent-properties-negative-exponent-definition.html}{\text{Negative Exponent Definition}}\\ \amp=\frac{3}{4^{\sfrac{1}{2}}\left(k^{\sfrac{2}{3}}\right)^{\sfrac{1}{2}}}\amp\amp\text{by the } \knowl{./knowl/xref/item-exponent-properties-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/xref/item-exponent-properties-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/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \amp=\frac{3}{2\sqrt[3]{k}} \end{align*}
Example 6.3.17.
Use the radical properties in List 14 to simplify the following expressions, answering with positive rational exponents only (no radicals).
- \(\displaystyle \sqrt[7]{128y^4}\)
- \(\displaystyle \frac{\sqrt[3]{64z^2}}{\sqrt[4]{z}}\)
- \(\displaystyle \frac{\sqrt{36x}}{\sqrt[5]{x^3}}\)
- \(\displaystyle \sqrt[3]{n^2}\sqrt[4]{n^3}\)
- \(\displaystyle \sqrt[7]{\sqrt[3]{x}}\)
Explanation.
- \begin{align*} \sqrt[7]{128y^4}\amp=(128y^4)^{\sfrac{1}{7}}\amp\amp\text{by the}~\knowl{./knowl/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \amp=(128)^{\sfrac{1}{7}}(y^4)^{\sfrac{1}{7}}\amp\amp\text{by the}~\knowl{./knowl/xref/item-exponent-properties-product-to-a-power.html}{\text{Product to a Power Rule}}\\ \amp=\sqrt[7]{128}(y^4)^{\sfrac{1}{7}}\amp\amp\text{by the}~\knowl{./knowl/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \amp=2y^{\sfrac{4}{7}}\amp\amp\text{by the}~\knowl{./knowl/xref/item-exponent-properties-power-to-a-power.html}{\text{Power to a Power Rule}} \end{align*}
- \begin{align*} \frac{\sqrt[3]{64z^2}}{\sqrt[4]{z}}\amp=\frac{(64z^2)^{\sfrac{1}{3}}}{z^{\sfrac{1}{4}}}\amp\amp\text{by the}~\knowl{./knowl/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \amp=\frac{(64)^{\sfrac{1}{3}}(z^2)^{\sfrac{1}{3}}}{z^{\sfrac{1}{4}}}\amp\amp\text{by the}~\knowl{./knowl/xref/item-exponent-properties-product-to-a-power.html}{\text{Product to a Power Rule}}\\ \amp=\frac{\sqrt[3]{64}(z^2)^{\sfrac{1}{3}}}{z^{\sfrac{1}{4}}}\amp\amp\text{by the}~\knowl{./knowl/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \amp=\frac{4z^{\sfrac{2}{3}}}{z^{\sfrac{1}{4}}}\amp\amp\text{by the}~\knowl{./knowl/xref/item-exponent-properties-power-to-a-power.html}{\text{Power to a Power Rule}}\\ \amp=4z^{\sfrac{2}{3}-\sfrac{1}{4}}\amp\amp\text{by the}~\knowl{./knowl/xref/item-exponent-properties-quotient.html}{\text{Quotient Rule}}\\ \amp=4z^{\sfrac{8}{12}-\sfrac{3}{12}}\\ \amp=4z^{\sfrac{5}{12}} \end{align*}
- \begin{align*} \frac{\sqrt{36x}}{\sqrt[5]{x^3}}\amp=\frac{(36x)^{\sfrac{1}{2}}}{x^{\sfrac{3}{5}}}\amp\amp\text{by the}~\knowl{./knowl/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}}\\ \amp=\frac{36^{\sfrac{1}{2}}x^{\sfrac{1}{2}}}{x^{\sfrac{3}{5}}}\amp\amp\text{by the}~\knowl{./knowl/xref/item-exponent-properties-product-to-a-power.html}{\text{Product to a Power Rule}}\\ \amp=\frac{\sqrt{36}x^{\sfrac{1}{2}}}{x^{\sfrac{3}{5}}}\amp\amp\text{by the}~\knowl{./knowl/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}}\\ \amp=6x^{\sfrac{1}{2}-\sfrac{3}{5}}\amp\amp\text{by the}~\knowl{./knowl/xref/item-exponent-properties-quotient.html}{\text{Quotient Rule}}\\ \amp=6x^{\sfrac{5}{10}-\sfrac{6}{10}}\\ \amp=6x^{\sfrac{-1}{10}}\\ \amp=\frac{6}{x^{\sfrac{1}{10}}}\amp\amp\text{by the}~\knowl{./knowl/xref/item-exponent-properties-negative-exponent-definition.html}{\text{Negative Exponent Definition}} \end{align*}
- \begin{align*} \sqrt[3]{n^2}\sqrt[4]{n^3}\amp=n^{\sfrac{2}{3}}n^{\sfrac{3}{4}}\amp\amp\text{by the}~\knowl{./knowl/xref/fact-rational-exponent-property-ii.html}{\text{Full Radicals and Rational Exponents Property}}\\ \amp=n^{\sfrac{2}{3}+\sfrac{3}{4}}\amp\amp\text{by the}~\knowl{./knowl/xref/item-exponent-properties-product.html}{\text{Product Rule}}\\ \amp=n^{\sfrac{8}{12}+\sfrac{9}{12}}\\ \amp=n^{\sfrac{17}{12}} \end{align*}
- \begin{align*} \sqrt[7]{\sqrt[3]{x}}\amp=\sqrt[7]{x^{\sfrac{1}{3}}}\amp\amp\text{by the}~\knowl{./knowl/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \amp=\left(x^{\sfrac{1}{3}}\right)^{\sfrac{1}{7}}\amp\amp\text{by the}~\knowl{./knowl/xref/fact-rational-exponent-property-i.html}{\text{Radicals and Rational Exponents Property}}\\ \amp=x^{\sfrac{1}{21}}\amp\amp\text{by the}~\knowl{./knowl/xref/item-exponent-properties-power-to-a-power.html}{\text{Power to a Power Rule}} \end{align*}
We will end a with a short application of rational exponents. Kepler’s Laws of Orbital 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.
2
en.wikipedia.org/wiki/Kepler%27s_laws_of_planetary_motionExample 6.3.18. 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 where \(G\) is about \(8.65\times 10^{-13}\) km3⁄kg·h2 and \(M_E\) stands for the mass of the Earth 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:
3
en.wikipedia.org/wiki/Gravitational_constant4
en.wikipedia.org/wiki/Earth_mass
\begin{equation*}
T\approx2.76\times 10^{-6}r^{\sfrac{3}{2}}
\end{equation*}
Most satellites orbit in what is called 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.
5
en.wikipedia.org/wiki/Low_Earth_orbitExplanation.
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
Exercise Group.
Use the properties of exponents to simplify the expression. If there are any exponents in your answer, they should be positive.
1.
\({x^{19}}\cdot{x^{20}}\)
2.
\({r^{2}}\cdot{r^{14}}\)
3.
\(\displaystyle{\left(x^{3}\right)^{5}}\)
4.
\(\displaystyle{\left(y^{4}\right)^{12}}\)
5.
\(\left(\displaystyle\frac{9x^{5}}{10}\right)^{3}\)
6.
\(\left(\displaystyle\frac{5x^{6}}{4}\right)^{2}\)
7.
\(\left(-4t^{8}\right)^3\)
8.
\(\left(-10r^{9}\right)^3\)
9.
\(\displaystyle\frac{{r^{17}}}{{r^{4}}}\)
10.
\(\displaystyle\frac{{t^{19}}}{{t^{16}}}\)
11.
\(x^{-12}\cdot x^{5}\)
12.
\(x^{-6}\cdot x^{5}\)
13.
\((-4y^{-17})\cdot (-9y^{4})\)
14.
\((-2y^{-11})\cdot (5y^{7})\)
Skills Practice
Exercise Group.
Evaluate each expression without help from a calculator.
15.
\(\displaystyle{ 81^{-\frac{1}{2}} }\)
16.
\(\displaystyle{ 27^{-\frac{2}{3}} }\)
17.
\(\displaystyle{ \left(\frac{1}{81}\right)^{-\frac{3}{4}} }\)
18.
\(\displaystyle{ \left(\frac{1}{9}\right)^{-\frac{3}{2}} }\)
19.
\(\displaystyle{ \sqrt[5]{32^{4}} }\)
20.
\(\displaystyle{ \sqrt[2]{25^{3}} }\)
21.
\(\displaystyle{ \sqrt[5]{1024} }\)
22.
\(\displaystyle{ \sqrt[3]{64} }\)
23.
\(\displaystyle{ \sqrt[3]{{-{\frac{8}{27}}}} }\)
24.
\(\displaystyle{ \sqrt[3]{{-{\frac{27}{125}}}} }\)
25.
\(\displaystyle{ \sqrt[3]{{-{\frac{27}{64}}}} }\)
26.
\(\displaystyle{ \sqrt[3]{{-{\frac{27}{125}}}} }\)
27.
(a)
\(\displaystyle{ 81^{\frac{1}{2}} }\)
(b)
\(\displaystyle{ (-81)^{\frac{1}{2}} }\)
(c)
\(\displaystyle{ -81^{\frac{1}{2}} }\)
28.
(a)
\(\displaystyle{ 100^{\frac{1}{2}} }\)
(b)
\(\displaystyle{ (-100)^{\frac{1}{2}} }\)
(c)
\(\displaystyle{ -100^{\frac{1}{2}} }\)
29.
(a)
\(\displaystyle{ 8^{\frac{1}{3}} }\)
(b)
\(\displaystyle{ (-8)^{\frac{1}{3}} }\)
(c)
\(\displaystyle{ -8^{\frac{1}{3}} }\)
30.
(a)
\(\displaystyle{ 27^{\frac{1}{3}} }\)
(b)
\(\displaystyle{ (-27)^{\frac{1}{3}} }\)
(c)
\(\displaystyle{ -27^{\frac{1}{3}} }\)
31.
(a)
\(\displaystyle{ \sqrt[3]{8} }\)
(b)
\(\displaystyle{ \sqrt[3]{-8} }\)
(c)
\(\displaystyle{ -\sqrt[3]{8} }\)
32.
(a)
\(\displaystyle{ \sqrt[3]{27} }\)
(b)
\(\displaystyle{ \sqrt[3]{-27} }\)
(c)
\(\displaystyle{ -\sqrt[3]{27} }\)
33.
(a)
\(\displaystyle{ \sqrt[4]{16} }\)
(b)
\(\displaystyle{ \sqrt[4]{-16} }\)
(c)
\(\displaystyle{ -\sqrt[4]{16} }\)
34.
(a)
\(\displaystyle{ \sqrt[4]{81} }\)
(b)
\(\displaystyle{ \sqrt[4]{-81} }\)
(c)
\(\displaystyle{ -\sqrt[4]{81} }\)
Exercise Group.
Use a calculator to approximate the expression with a decimal to four significant digits.
35.
\(\displaystyle{\sqrt[5]{6^{2}}}\)
36.
\(\displaystyle{\sqrt[7]{17^{4}}}\)
37.
\(\displaystyle{\sqrt[4]{19^{3}}}\)
38.
\(\displaystyle{\sqrt[3]{9^{2}}}\)
Convert Radicals to Fractional Exponents.
Write the expression with rational exponents.
39.
\(\displaystyle{\sqrt[8]{y}}\)
40.
\(\displaystyle{\sqrt[5]{z}}\)
41.
\(\displaystyle{\sqrt[4]{2 t + 9}}\)
42.
\(\displaystyle{\sqrt[3]{8 r + 3}}\)
43.
\(\displaystyle{ \sqrt[5]{m} }\)
44.
\(\displaystyle{ \sqrt{n} }\)
45.
\(\displaystyle{ \frac{1}{\sqrt[7]{b^{4}}} }\)
46.
\(\displaystyle{ \frac{1}{\sqrt[4]{c^{5}}} }\)
Convert Fractional Exponents to Radicals.
Convert the expression to use radical notation.
47.
\(\displaystyle{{x^{\frac{2}{3}}}}\)
48.
\(\displaystyle{{y^{\frac{4}{5}}}}\)
49.
\(\displaystyle{{z^{\frac{4}{3}}}}\)
50.
\(\displaystyle{{r^{\frac{2}{3}}}}\)
51.
\(\displaystyle{{2^{\frac{1}{5}}r^{\frac{4}{5}}}}\)
52.
\(\displaystyle{{10^{\frac{1}{4}}m^{\frac{3}{4}}}}\)
53.
\(n^{\frac{5}{8}}\)
54.
\(b^{\frac{5}{8}}\)
55.
\(c^{-\frac{2}{7}}\)
56.
\(x^{-\frac{5}{6}}\)
57.
\(2^{\frac{1}{7}} y^{\frac{4}{7}}\)
58.
\(2^{\frac{1}{8}} z^{\frac{5}{8}}\)
Simplifying Expressions with Rational Exponents.
Simplify the expression, answering with rational exponents and not radicals.
59.
\(\displaystyle{\sqrt[11]{t}\,\sqrt[11]{t}}\)
60.
\(\displaystyle{\sqrt[7]{r}\,\sqrt[7]{r}}\)
61.
\(\displaystyle{\sqrt{4 m}}\)
62.
\(\displaystyle{\sqrt[5]{32 n^{4}}}\)
63.
\(\displaystyle{\frac{\sqrt{25 a}}{\sqrt[10]{a^{3}}}}\)
64.
\(\displaystyle{\frac{\sqrt{100 c}}{\sqrt[6]{c^{5}}}}\)
65.
\(\displaystyle{\frac{\sqrt[3]{27 x^{4}}}{\sqrt[6]{x^{5}}}}\)
66.
\(\displaystyle{\frac{\sqrt[3]{27 y^{4}}}{\sqrt[6]{y^{5}}}}\)
67.
\(\displaystyle{\sqrt[5]{z}\cdot\sqrt[10]{z^{3}}}\)
68.
\(\displaystyle{\sqrt{t} \cdot \sqrt[6]{t^{5}}}\)
69.
\(\displaystyle{\sqrt[4]{\sqrt[3]{r}}}\)
70.
\(\displaystyle{\sqrt[5]{\sqrt{m}}}\)
71.
\(\displaystyle{\sqrt{n}\sqrt[8]{n}}\)
72.
\(\displaystyle{\sqrt{t}\sqrt[9]{t}}\)
Applications
Exercise Group.
On a guitar, there are \(12\) frets separating a note and the same note one octave higher.
73.
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 a small denominator. Two notes with this frequency ratio form a “perfect fifth” in music. What is that fraction?
74.
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 a small denominator. Two notes with this frequency ratio form a “major third” in music. What is that fraction?
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