Question

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

$$\begin{gathered} a.\sqrt{\frac{x^{10}}{x^6}} \\ b.\sqrt[3]{\frac{p^{11}}{p^2}} \\ c.\sqrt[4]{\frac{q^{17}}{q^{13}}} \end{gathered}$$

Short answer

The simplified expressions are,

$$\begin{gathered} a.x^2. \\ b.\left|p^3\right|. \\ c.\left|q\right|. \end{gathered}$$

Step-by-step solution

Step 1 Given expressions are,

$$\begin{gathered} a.\sqrt{\frac{x^{10}}{x^6}} \\ b.\sqrt[3]{\frac{p^{11}}{p^2}} \\ c.\sqrt[4]{\frac{q^{17}}{q^{13}}} \end{gathered}$$

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 src="data:image/svg+xml;base64,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" localid="1645431234704" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/05b8b9f2-00cd-4e3a-98cd-12ed81fac7d3.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T155621Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=617745363a10bccdbf4ec2f9b1178788ca009712551f0250b3d22a7c4467813c" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/05b8b9f2-00cd-4e3a-98cd-12ed81fac7d3.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220221%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220221T081523Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=4840fbf3d88edc305e00be411d59e0df04ffbafcbbae143c0e7d527e319949ed" x10x6

Now use the quotient property.
$$\begin{gathered} \sqrt{\frac{x^{10}}{x^6}}=\sqrt{x^{10-6}} \\ =\sqrt{x^4} \\ =x^2 \end{gathered}$$

Part b Step 2p11p23

Now use the quotient property.

$$\begin{gathered} \sqrt[3]{\frac{p^{11}}{p^2}}=\sqrt[3]{p^{11-2}} \\ =\sqrt[3]{p^9} \\ =\left|p^3\right| \end{gathered}$$

Part c Step 2q17q134

Now use the quotient property.