Q. 114 - Intermediate Algebra

Question

Use the Quotient Property to simplify square roots.

(a) $\frac{\sqrt{72n^{11}}}{\sqrt{2n}}$

(b) $\sqrt[3]{\frac{162}{6}}$

(c) $\sqrt[4]{\frac{160r^{10}}{5r^3}}$

Step-by-Step Solution

Answer

(a) $6\left|n^5\right|$

(b) $3$

(c) $3\left|r\right|\sqrt[4]{2r^3}$

Part (a) Step 1. Given information.

Given expression:

$\frac{\sqrt{72n^{11}}}{\sqrt{2n}}$.

Part (a) Step 2. Write the concept.

For simplifying the radical expression following formula is used:

$$\begin{gathered} \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}} and \\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}} \end{gathered}$$

if $\sqrt[n]{a} and \sqrt[n]{b}$ are the real number, $b\ne 0$for any integer $n\ge 2$.

Part (a) Step 3. Calculation

$$\begin{gathered} \frac{\sqrt{72n^{11}}}{\sqrt{2n}}=\sqrt{\frac{72n^{11}}{2n}} \\ =\sqrt{\frac{36.2.n^{10}.n}{2.n}} \\ =\sqrt{36.n^{10}} \\ =\sqrt{36}.\sqrt{n^{10}} \\ =\sqrt{6^2.{\left(n^5\right)}^2} \\ =6\left|n^5\right| \end{gathered}$$

Part (b) Step 1. Given information.

Given radical expression:

$\sqrt[3]{\frac{162}{6}}$.

Part (b) Step 2. Write the concept.

For simplifying the radical expression following formula is used:

$$\begin{gathered} \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}} and \\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}} \end{gathered}$$

if $\sqrt[n]{a} and \sqrt[n]{b}$ are the real number, $b\ne 0$for any integer $n\ge 2$.

Part (b) Step 3. Calculation

$$\begin{gathered} \sqrt[3]{\frac{162}{3}}=\sqrt[3]{\frac{27.6}{6}} \\ =\sqrt[3]{27} \\ =3 \end{gathered}$$

Part (c) Step 1. Given information.

Given expression:

$\sqrt[4]{\frac{160r^{10}}{5r^3}}$.

Part (c) Step 2. Write the concept.

radical expression following formula is used:

$$\begin{gathered} \sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}} and \\ \frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}} \end{gathered}$$

if $\sqrt[n]{a} and \sqrt[n]{b}$ are the real number, $b\ne 0$for any integer $n\ge 2$.

Part (c) Step 3. Calculation

$$\begin{gathered} \sqrt[4]{\frac{160r^{10}}{5r^3}}=\sqrt[4]{\frac{16.2.5.r^{10}}{5.r^3}} \\ =\sqrt[4]{16.2.r^2} \\ =\sqrt[4]{2^4.2r^4.y^3} \\ =3\left|r\right|\sqrt[4]{2r^3} \end{gathered}$$