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Section 3.4 Rational Exponents

Subsection 1. Perform operations on fractions

When working with rational exponents, we will need to perform operations on fractions.

Subsubsection Examples

Example 3.43.

Add \(~\dfrac{-3}{4}+\left(\dfrac{-5}{8}\right)\)

Solution

The LCD for the fractions is 8, so we build the first fraction:

\begin{equation*} \dfrac{-3}{4} \cdot \alert{\dfrac{2}{2}} = \dfrac{-6}{8} \end{equation*}

Then we combine like fractions:

\begin{equation*} \dfrac{-6}{8}+\left(\dfrac{-5}{8}\right) = \dfrac{-6+(-5)}{8} = \dfrac{-11}{8} \end{equation*}
Example 3.44.

Subtract \(~\dfrac{-5}{6}-\left(\dfrac{-3}{4}\right)\)

Solution

The LCD for the fractions is 12, so we build each fraction:

\begin{equation*} \dfrac{-5}{6} \cdot \alert{\dfrac{2}{2}} = \dfrac{-10}{12};~~~~\dfrac{-3}{4} \cdot \alert{\dfrac{3}{3}} = \dfrac{-9}{12} \end{equation*}

Then we combine like fractions:

\begin{equation*} \dfrac{-10}{12}-\left(\dfrac{-9}{12}\right) = \dfrac{-10+9}{12} = \dfrac{-1}{12} \end{equation*}
Example 3.45.

Multiply \(~\dfrac{-2}{3}\left(\dfrac{5}{4}\right)\)

Solution

We multiply numerators together, and multiply denominators together:

\begin{equation*} \dfrac{-2}{3}\left(\dfrac{5}{4}\right) = \dfrac{-2 \cdot 5}{3 \cdot 4} = \dfrac{-10}{12} \end{equation*}

Then we reduce:

\begin{equation*} \dfrac{-10}{12} = \dfrac{-5 \cdot \cancel{2}}{6 \cdot \cancel{2}} = \dfrac{-5}{6} \end{equation*}

Subsubsection Exercises

Add \(~\dfrac{-3}{4}+\dfrac{1}{3}\)

Answer

\(\dfrac{-5}{12}\)

Subtract \(~\dfrac{3}{8}-\left(\dfrac{-1}{6}\right)\)

Answer

\(\dfrac{13}{24}\)

Multiply \(~\dfrac{3}{8} \cdot \left(\dfrac{-1}{6}\right)\)

Answer

\(\dfrac{-1}{16}\)

Subsection 2. Convert between fractions and decimals

Rational exponents may also be written in decimal form.

Subsubsection Examples

Example 3.49.

Convert \(~0.016~\) to a common fraction.

Solution

The numerator of the fraction is 016, or 16. The last digit, 6, is in the thousandths place, so the denominator of the fraction is 1000. Thus, \(0.016=\dfrac{16}{1000}\text{.}\) We can reduce this fraction by dividing top and bottom by 8:

\begin{equation*} ~ \dfrac{16}{1000} = \dfrac{\cancel{8} \cdot 2}{\cancel{8} \cdot 125} = \dfrac{2}{125} \end{equation*}
Example 3.50.

Convert \(~\dfrac{5}{16}~\) to a decimal fraction.

Solution

Using a calculator, divide 5 by 16:

\(\qquad\qquad 5\) ÷ \(16 = 0.3125\)

Example 3.51.

Convert \(~\dfrac{5}{11}~\) to a decimal fraction.

Solution

Using a calculator, divide 5 by 11:

\(\qquad\qquad 5\) ÷ \(11= 0.45454545 ... \)

This is a nonterminating decimal, which we indicate by a repeater bar:

\begin{equation*} \dfrac{5}{11}= 0.45454545 ... = 0.\overline{45} \end{equation*}

Subsubsection Exercises

Convert \(~0.1062~\) to a common fraction.

Answer
\(\dfrac{531}{5000}\)

Convert \(~2.08~\) to a common fraction.

Answer
\(\dfrac{52}{25}\)

Convert \(~\dfrac{4}{15}~\) to a decimal fraction.

Answer
\(0.2\overline{6}\)

Subsection 3. Solve equations

To solve an equation of the form \(~x^n = k\text{,}\) we can raise both sides to the reciprocal of the exponent:

\begin{align*} (x^n)^{1/n} \amp = k^{1/n}\\ x \amp = k^{1/n} \end{align*}

because \(~(x^n)^{1/n} = x^{n(1/n)} = x^1\text{.}\)

Subsubsection Examples

Example 3.55.

Solve \(~0.6x^4 = 578\text{.}\) Round your answer to hundredths.

Solution

First, we isolate the power.

\begin{align*} 0.6x^4 \amp = 578 \amp\amp \blert{\text{Divide both sides by 0.6.}}\\ x^4 \amp = 963.\overline{3} \end{align*}

We raise both sides to the reciprocal of the power.

\begin{align*} (x^4)^{1/4} \amp = (963.\overline{3})^{1/4} \amp\amp \blert{\text{By the third law of exponents,}~ (x^4)^{1/4}=x.}\\ x \amp = 5.57 \end{align*}

To evaluate \((963.\overline{3})^{1/4}\text{,}\) enter \(~~\text{ANS}\)^ \(.25\) ENTER

Example 3.56.

Solve \(~x^{2/3}-4=60\text{.}\)

Solution

First, we isolate the power.

\begin{align*} x^{2/3}-4 \amp = 60 \amp\amp \blert{\text{Add 4 to both sides.}}\\ x^{2/3} \amp = 64 \end{align*}

We raise both sides to the reciprocal of the power.

\begin{align*} \left(x^{2/3}\right)^{3/2} \amp = 64^{3/2} \amp\amp \blert{64^{3/2}=\left(64^{1/2}\right)^3=8^3}\\ x \amp = 512 \end{align*}

Or we can evaluate \(~64^{3/2}~\) by entering \(~~64\) ^ \(1.5\) ENTER

Example 3.57.

Solve \(~18x^{0.24} = 6.5\text{.}\) Round your answer to thousandths.

Solution

First, we isolate the power.

\begin{align*} 18x^{0.24} \amp = 6.5 \amp\amp \blert{\text{Divide both sides by 18.}}\\ x^{0.24} \amp = 0.36\overline{1} \end{align*}

We raise both sides to the reciprocal of the power.

\begin{align*} \left(x^{0.24}\right)^{1/0.24} \amp = (0.36\overline{1})^{1/0.24}\\ x \amp = 0.014 \end{align*}

We evaluate \((0.36\overline{1})^{1/0.24}\) by entering \(~~\text{ANS}\) ^ ( \(1\) ÷ \(.24\) ) ENTER

Subsubsection Exercises

Solve \(~4x^5 = 1825~\text{.}\) Round your answer to thousandths.

Answer
\(3.403\)

Solve \(~\dfrac{3}{4}x^{3/4} = 36~\text{.}\) Round your answer to thousandths.

Answer
\(174.444\)

Solve \(~0.2x^{1.4}+1.8=12.3~\text{.}\) Round your answer to thousandths.

Answer
\(16.931\)