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Quantitative Skills and Reasoning Activity Workbook

Megan Patnott

Section 16.1 Roots and Fractional Powers Prep Activities

We learned in Section 12.2 that cube roots and the one-third power are the same, i.e., that \(\sqrt[3]{x} = x^{1/3}\text{.}\) In fact, this is a special case of the general rule that nth roots and the 1/n power are the same, i.e.,
\begin{equation*} \sqrt[n]{x} = x^{1/n}\text{.} \end{equation*}
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Note: We use square roots so often that we don’t usually bother to write the number on the root. But, \(\sqrt{x} = \sqrt[2]{x}\text{,}\) so \(\sqrt{x} = x^{1/2}\text{.}\)
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Prep Activity 16.1.1.

Using the Product Rule for Exponents,
\begin{equation*} x^{1/2}x^{1/2} = x^1 \qquad \text{ and } \qquad x^{1/3}x^{1/3}x^{1/3} = x^1\text{.} \end{equation*}
Why does this observation support the reasonableness of the rules that \(x^{1/2} = \sqrt{x}\) and \(x^{1/3} = \sqrt[3]{x}\text{?}\)
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Prep Activity 16.1.2.

Let’s practice using our new rule, along with other exponent rules, to rewrite expressions. Remember, if you’re not sure what to do, look for an opportunity to use any one of the exponent rules, use it, and then re-evaluate.
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(a)

Rewrite without a root symbol: \(\sqrt[4]{a}\)
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(b)

Rewrite with a root symbol: \(\left(17b\right)^{1/5}\)
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(c)

Rewrite as \((\text{number}) \times c^{\text{power}}\text{:}\) \(\frac13 \cdot c^3 \cdot \sqrt[4]{81c}\)
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(d)

Rewrite as \((\text{number}) \times d^{\text{power}}\text{:}\) \(\left(\sqrt[5]{32d}\right)^3\)
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(e)

Rewrite with a root symbol:\(\frac{\left(e^{1/2}\right)^3}{e^{1/2}}\)
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Prep Activity 16.1.3.

We introduced the one-third power in Section 12.2 to help us solve the equation
\begin{equation*} 10 = x^3\text{.} \end{equation*}
Notice that the Power Rule for Exponents tells us that
\begin{equation*} \left(x^n\right)^{1/n} = x \qquad \text{ and } \qquad \left(x^{1/n}\right)^n = x\text{.} \end{equation*}
That is, the nth power and the 1/n power undo each other. Use this fact to find all solutions to each of the equations below.
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Prep Activity 16.1.4.

You’ll need to be able to do the following things for this lesson. Rate how confident you are on a scale of 1 - 5 (1 = not confident and 5 = very confident).
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Table 16.1.1.
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Skill or Concept: I can … Rating from 1 to 5
Use the equality \(\sqrt[n]{x} = x^{1/n}\) to rewrite expressions.
Solve equations using the inverse relationship between \(x^n\) and \(x^{1/n}\text{.}\)
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