Question

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

$$\begin{gathered} a.\sqrt{\frac{p^{20}}{p^{10}}} \\ b.\sqrt[5]{\frac{d^{12}}{d^7}} \\ c.\sqrt[8]{\frac{m^{12}}{m^4}} \end{gathered}$$

Short answer

The simplified expressions are,

$$\begin{gathered} a.\left|p^5\right|. \\ b.\left|d\right|. \\ c.\left|m\right|. \end{gathered}$$

Step-by-step solution

Step 1 Given expressions are,

$$\begin{gathered} a.\sqrt{\frac{p^{20}}{p^{10}}} \\ b.\sqrt[5]{\frac{d^{12}}{d^7}} \\ c.\sqrt[8]{\frac{m^{12}}{m^4}} \end{gathered}$$

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

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

Part a Step 2p20p10 

Now use the quotient property.

$$\begin{gathered} \sqrt{\frac{p^{20}}{p^{10}}}=\sqrt{p^{20-10}} \\ =\sqrt{p^{10}} \\ =\left|p^5\right| \end{gathered}$$

Part b Step 2 d12d75  

Now use the quotient property.
$$\begin{gathered} \sqrt[5]{\frac{d^{12}}{d^7}}=\sqrt[5]{d^{12-7}} \\ =\sqrt[5]{d^5} \\ =\left|d\right| \end{gathered}$$

Part c Step 2  m12m48

Now use the quotient property.
$$\begin{gathered} \sqrt[8]{\frac{m^{12}}{m^4}}=\sqrt[8]{m^{12-4}} \\ =\sqrt[8]{m^8} \\ =\left|m\right| \end{gathered}$$