Section4.5Radical Expressions and Rational Exponents
Recall that in Subsection 4.3.3, 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. Section 1 covers an introduction to exponent properties, and there is more in Section 2. The basic properties are summarized in List 4.2.13. These properties are still true and we can use them throughout this section whenever they might help.
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.
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 2, but you should confirm this.
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 \(1.817\text{.}\)
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*}
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.
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*}
\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*}
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.
Additionally, if \(n\) is an odd natural number, then even when \(a\) is negative, we still have \(a^{\sfrac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m\text{.}\)
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.
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).
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.
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.
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.
\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*}
\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*}
\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*}
\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*}
\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*}
\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.
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:
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:
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.
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.
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?
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?
