Q. 106 - Intermediate Algebra

Question

Use the Quotient Property to simplify square roots.

(a) $\sqrt{\frac{45r^3}{s^{10}}}$

(b) $\sqrt[3]{\frac{625u^{10}}{v^3}}$

(c) $\sqrt[4]{\frac{729c^{21}}{d^8}}$

Step-by-Step Solution

Answer

(a) $\frac{3\left|r\right|\sqrt{5r}}{\left|s^5\right|}$

(b) $\frac{5u^3\sqrt[3]{25u}}{v}$

(c) $\frac{3\left|c^5\right|\sqrt[4]{9c}}{d^2}$

Part (a) Step 1. Given information.

Given expression:

$\sqrt{\frac{45r^3}{s^{10}}}$.

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

$$\begin{gathered} \sqrt{\frac{45r^3}{s^{10}}}=\sqrt{\frac{9r^2.5r}{s^{10}}} \\ =\frac{\sqrt{9r^2.5r}}{\sqrt{s^{10}}} \\ =\frac{3\left|r\right|\sqrt{5r}}{\left|s^5\right|} \end{gathered}$$

Part (b) Step 1. Given information.

Given radical expression:

$\sqrt[3]{\frac{625u^{10}}{v^3}}$.

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

$$\begin{gathered} \sqrt[3]{\frac{625u^{10}}{v^3}}=\sqrt[3]{\frac{25u^9.25u}{v^3}} \\ =\frac{\sqrt[3]{25u^9.25u}}{\sqrt[3]{v^3}} \\ =\frac{5u^3\sqrt[3]{25u}}{v} \end{gathered}$$

Part (c) Step 1. Given information.

Given expression:

$\sqrt[4]{\frac{729c^{21}}{d^8}}$.

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

$$\begin{gathered} \sqrt[4]{\frac{729c^{21}}{d^8}}=\sqrt[4]{\frac{81c^{20}.9c}{d^8}} \\ =\frac{\sqrt[4]{81c^{20}.9c}}{\sqrt[4]{d^8}} \\ =\frac{3\left|c^5\right|\sqrt[4]{9c}}{\sqrt[4]{\left|d^2\right|}} \end{gathered}$$