Lane ORCCA (2020–2021): Open Resources for Community College Algebra

1. \(\displaystyle \begin{aligned}[t] 4^{\sfrac{1}{2}}\amp=\sqrt{4}\\ \amp=2 \end{aligned}\)

2. \(\begin{aligned}[t] (-9)^{\sfrac{1}{2}}\amp=\sqrt{-9} \end{aligned}\)This value is non-real.

3. Without parentheses around \(-16\text{,}\) the negative sign in this problem should be left out of the radical.

   \(\begin{aligned}[t] -16^{\sfrac{1}{4}}\amp=-\sqrt[4]{16}\\ \amp=-2 \end{aligned}\)

4. \(\displaystyle \begin{aligned}[t] 64^{-\sfrac{1}{3}}\amp=\frac{1}{64^{\sfrac{1}{3}}}\\ \amp=\frac{1}{\sqrt[3]{64}}\\ \amp=\frac{1}{4} \end{aligned}\)

5. \(\displaystyle \begin{aligned}[t] (-27)^{\sfrac{1}{3}}\amp=\sqrt[3]{-27}\\ \amp=-3 \end{aligned}\)

6. \(\displaystyle \begin{aligned}[t] 3^{\sfrac{1}{2}}\cdot3^{\sfrac{1}{2}}\amp=\sqrt{3}\cdot\sqrt{3}\\ \amp=\sqrt{3\cdot 3}\\ \amp=\sqrt{9}\\ \amp=3 \end{aligned}\)

7. \(\displaystyle \begin{aligned}[t] 12^0\amp=1 \end{aligned}\)

1. 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*}

2. 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 could choose to simplify our answer in a different way. Note that neither one is technically preferred over the other except that perhaps the first way is simpler.

   \begin{align*} (5x)^{\sfrac{1}{3}}\amp=5^{\sfrac{1}{3}}x^{\sfrac{1}{3}} \amp\amp\text{by the }\knowl{./knowl/item-exponent-rules-product-to-a-power.html}{\text{Product to a Power Rule}}\\ \amp=\sqrt[3]{5}\sqrt[3]{x} \amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}} \end{align*}

3. 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=(-27)^{\sfrac{1}{3}}\cdot \left(x^{12}\right)^{\sfrac{1}{3}}\amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-product-to-a-power.html}{\text{Product to a Power Rule}}\\ \amp=(-27)^{\sfrac{1}{3}}\cdot \left(x^{12\cdot\sfrac{1}{3}}\right) \amp\amp\text{by the } \knowl{./knowl/item-exponent-rules-power-to-a-power.html}{\text{Power to a Power Rule}}\\ \amp=\sqrt[3]{-27}\cdot x^4 \amp\amp\text{by the } \knowl{./knowl/fact-rational-exponent-rule-i.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=-3x^4 \end{align*}

4. 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*}

1. The expression \(27^{\sfrac{4}{3}}\) can be evaluated in the following two ways by Fact 13.2.9.

   \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{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{Radicals and Rational Exponents Rule}}\\ \amp=3^4\\ \amp=81 \end{align*}

   The calculations 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 13.2.10.

2. The expression \(x^{\sfrac{4}{3}}\) can be evaluated in the following two ways by Fact 13.2.9.

   \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{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{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 13.2.10.

1. \begin{align*} 8^{\sfrac{2}{3}}\amp=\left(\sqrt[3]{8}\right)^2\amp\amp\text{by the second part of the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=2^2\\ \amp=4 \end{align*}
2. \begin{align*} 16^{-\sfrac{3}{2}}\amp=\frac{1}{16^{\sfrac{3}{2}}}\amp\amp\text{by the }\knowl{./knowl/item-exponent-rules-negative-exponent-definition.html}{\text{Negative Exponent Rule}}\\ \amp=\frac{1}{\left(\sqrt{16}\right)^3}\amp\amp\text{by the second part of the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=\frac{1}{4^3}\\ \amp=\frac{1}{64} \end{align*}
3. \begin{align*} -16^{\sfrac{3}{4}}\amp=-\left(\sqrt[4]{16}\right)^3 \amp\amp\text{by the second part of the } \knowl{./knowl/fact-rational-exponent-rule-ii.html}{\text{Radicals and Rational Exponents Rule}}\\ \amp=-2^3\\ \amp=-8 \end{align*}

4. 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{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*}

1. \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{Radicals and Rational Exponents Rule}} \end{align*}
2. \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{Radicals and Rational Exponents Rule}}\\ \amp=\frac{1}{2\sqrt{y}} \end{align*}
3. \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{Radicals and Rational Exponents Rule}}\\ \amp=3^2\cdot\sqrt[3]{b^2}\\ \amp=9\sqrt[3]{b^2} \end{align*}
4. \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{Radicals and Rational Exponents Rule}}\\ \amp=(-2)^5\cdot p^{10}\\ \amp=-32p^{10} \end{align*}
5. \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{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{Radicals and Rational Exponents Rule}} \end{align*}
6. \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*}
7. \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*}
8. \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*}
9. \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*}
10. \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}\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*}

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.

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

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.