Question

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

$$\begin{gathered} a.\sqrt{\frac{y^4}{y^8}} \\ b.\sqrt[5]{\frac{u^{21}}{u^{11}}} \\ c.\sqrt[6]{\frac{v^{30}}{v^{12}}} \end{gathered}$$

Short answer

The simplified expressions are,

$$\begin{gathered} a.\frac{1}{y^2} \\ b.u^2 \\ c.\left|v^3\right| \end{gathered}$$

Step-by-step solution

Step 1 Given expressions are,

$$\begin{gathered} a.\sqrt{\frac{y^4}{y^8}} \\ b.\sqrt[5]{\frac{u^{21}}{u^{11}}} \\ c.\sqrt[6]{\frac{v^{30}}{v^{12}}} \end{gathered}$$

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   y4y8

Now use the quotient property.

$$\begin{gathered} \sqrt{\frac{y^4}{y^8}}=\sqrt{y^{4-8}} \\ =\sqrt{y^{-4}} \\ =\sqrt{\frac{1}{y^4}} \\ =\frac{1}{y^2} \end{gathered}$$

Part b Step 2  u21u115   

Now use the quotient property.

$$\begin{gathered} \sqrt[5]{\frac{u^{21}}{u^{11}}}=\sqrt[5]{u^{21-11}} \\ =\sqrt[5]{u^{10}} \\ =u^2 \end{gathered}$$

Part c Step 2    v30v126 

Now use the quotient property.

$$\begin{gathered} \sqrt[6]{\frac{v^{30}}{v^{12}}}=\sqrt[6]{v^{30-12}} \\ =\sqrt[6]{v^{18}} \\ =\left|v^3\right| \end{gathered}$$