Q. 94 - Intermediate Algebra

Question

In the following exercises, use the Quotient Property to simplify square roots.

$$\begin{gathered} a. \sqrt{\frac{q^8}{q^{14}}} \\ b. \sqrt[3]{\frac{r^{14}}{r^5}} \\ c. \sqrt[4]{\frac{c^{21}}{c^9}} \end{gathered}$$

Step-by-Step Solution

Answer

The simplified expressions are,

$$\begin{gathered} a. \left|\frac{1}{q^3}\right| \\ b. \left|r^3\right| \\ c. \left|c^3\right| \end{gathered}$$

Step 1 Given expressions are,

$$\begin{gathered} a. \sqrt{\frac{q^8}{q^{14}}} \\ b. \sqrt[3]{\frac{r^{14}}{r^5}} \\ c. \sqrt[4]{\frac{c^{21}}{c^9}} \end{gathered}$$

The objective is, we need to simplify the given expressions.

The quotient property to simplify under the radical is, $\frac{a^m}{a^n}=a^{m-n}$

Also we know that, $\sqrt[n]{a^n}=a$.

Part a Step 2         q 8 q 14    

Now use the quotient property.

$$\begin{gathered} \sqrt{\frac{q^8}{q^{14}}}=\sqrt{q^{8-14}} \\ =\sqrt{q^{-6}} \\ =\sqrt{\frac{1}{q^6}} \\ =\left|\frac{1}{q^3}\right| \end{gathered}$$

Part b Step 2           r 14 r 5 3

Now use the quotient property.

$$\begin{gathered} \sqrt[3]{\frac{r^{14}}{r^5}}=\sqrt[3]{r^{14-5}} \\ =\sqrt[3]{r^9} \\ =\sqrt[3]{{\left(r^3\right)}^3} \\ =\left|r^3\right| \end{gathered}$$

Part c Step 2           c 21 c 9 4  

Now use the quotient property.

$$\begin{gathered} \sqrt[4]{\frac{c^{21}}{c^9}}=\sqrt[4]{c^{21-9}} \\ =\sqrt[4]{c^{12}} \\ =\sqrt[4]{{\left(c^3\right)}^4} \\ =\left|c^3\right| \end{gathered}$$