Q. 8.96 - Intermediate Algebra

Question

Simplify:

(a) $\frac{\sqrt{300m^3{n}^7}}{\sqrt{3m^{5}n}}$

(b) $\frac{\sqrt[3]{-81p{q}^{-1}}}{\sqrt[3]{3p^{-2}{q}^5}}$

Step-by-Step Solution

Answer

The simplification of $\frac{\sqrt{300m^3{n}^7}}{\sqrt{3m^{5}n}}$ is $\frac{10n^3}{m^2}$

The simplification of $\frac{\sqrt[3]{-81p{q}^{-1}}}{\sqrt[3]{3p^{-2}{q}^5}}$ is $\frac{-3p}{q^2}$

Part (a) Step 1. Given

An expression $\frac{\sqrt{300m^3{n}^7}}{\sqrt{3m^{5}n}}$

To simplify the expression 

Part (a) Step 2. Rewrite using the quotient property

Rewrite using the quotient property 

$\frac{\sqrt{300m^3{n}^7}}{\sqrt{3m^{5}n}}=\sqrt{\frac{300m^3{n}^7}{3m^{5}n}}$

Remove the common factors. 

                      $=\sqrt{\frac{100n^6}{m^2}}$

Part (a) Step 3. Simplify the expression

$\frac{\sqrt{300m^3{n}^7}}{\sqrt{3m^{5}n}}=\sqrt{\frac{{\left(10n^3\right)}^2}{m^2}}$

                     $=\frac{10n^3}{m^2}$

Part (b) Step 1. Given

An expression  $\frac{\sqrt[3]{-81p{q}^{-1}}}{\sqrt[3]{3p^{-2}{q}^5}}$

To simplify the expression

Part (b) Step 2. Rewrite using the quotient property

Rewrite using the quotient property 

$\frac{\sqrt[3]{-81p{q}^{-1}}}{\sqrt[3]{3p^{-2}{q}^5}}=\sqrt[3]{\frac{-81p{q}^{-1}}{3p^{-2}{q}^5}}$

Remove the common factors. 

                      $=\sqrt[3]{\frac{-27p^3}{q^6}}$

Part (b) Step 3. Simplify the expression

$\frac{\sqrt[3]{-81p{q}^{-1}}}{\sqrt[3]{3p^{-2}{q}^5}}=\sqrt[3]{\frac{-27p^3}{q^6}}$

                      $=\sqrt[3]{\frac{{(-3)}^3{p}^3}{{\left(q^2\right)}^3}}$

                      $=\frac{-3p}{q^2}$

The simplification of the expression is $\frac{-3p}{q^2}$