Question

Use the Quotient Property to simplify square roots.

(a) $\sqrt{\frac{28p^7}{q^2}}$.

(b) $\sqrt[3]{\frac{81s^8}{t^3}}$.

(c)$\sqrt[4]{\frac{64p^{15}}{q^{12}}}$

Short answer

(a) $\frac{2\left|p^3\right|\sqrt{7x}}{\left|q\right|}$

(b) $\frac{3s^2\sqrt[3]{3s^2}}{t}$

(c)$\frac{2\left|p^3\right|\sqrt[4]{4p^3}}{\left|q^3\right|}$

Step-by-step solution

Part (a) Step 1. Given information.

The radical expression as:

$\sqrt{\frac{28p^7}{q^2}}$.

Part (a) Step 2. Write the concept.

To determine a simplified form of a 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 real numbers, $b\ne 0$for any integer $n\ge 2$.

Part (b) Step 3. Calculation.

Simplifying the expression.

$$\begin{gathered} \sqrt{\frac{28p^7}{q^2}}=\sqrt{\frac{4p^6.7p}{q^2}} \\ =\frac{\sqrt{4p^6.7p}}{\sqrt{q^2}} \\ =\frac{2\left|p^3\right|\sqrt{7p}}{\left|q\right|} \end{gathered}$$

Part (b) Step 1. Given information.

The radical expression as:

$\sqrt[3]{\frac{81s^8}{t^3}}$.

Part (b) Step 2. Write the concept.

To determine a simplified form of a 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 real numbers, $b\ne 0$for any integer $n\ge 2$.

Part (b) Step 3. Calculation.

Simplifying the expression.

$$\begin{gathered} \sqrt[3]{\frac{81s^8}{t^3}}=\sqrt[3]{\frac{27s^6.3s^2}{t^3}} \\ =\frac{\sqrt[3]{27s^6.3s^2}}{\sqrt[3]{t^3}} \\ =\frac{3s^2\sqrt[3]{3s^2}}{\left|t\right|} \end{gathered}$$

Part (c) Step 1. Given information.

The radical expression as:

$\sqrt[4]{\frac{64p^{15}}{q^{12}}}$.

Part (c) Step 2. Write the concept.

To determine a simplified form of a 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 real numbers, $b\ne 0$for any integer $n\ge 2$.

Part (c) Step 3. Calculation.

$$\begin{gathered} \sqrt[4]{\frac{64p^{15}}{q^{12}}}=\sqrt[4]{\frac{16p^{12}.4p^3}{q^{12}}} \\ =\frac{\sqrt[4]{16p^{12}.4p^3}}{\sqrt[4]{q^{12}}} \\ =\frac{\sqrt[4]{{\left(2p^3\right)}^4}.\sqrt[4]{4p^3}}{\sqrt[4]{{\left(q^3\right)}^4}} \\ =\frac{2\left|p^3\right|\sqrt[4]{4p^3}}{\left|q^3\right|} \end{gathered}$$