Q. 8.101 - Intermediate Algebra

Question

In the following exercise ,simplify 

$$\begin{gathered} \left(a\right) \frac{1}{\sqrt[3]{7}} \\ \left(b\right) \sqrt[3]{\frac{5}{12}} \\ \left(c\right) \frac{5}{\sqrt[3]{9y}} \end{gathered}$$

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right) \frac{\sqrt[3]{49}}{7} \\ \left(b\right) \frac{2\sqrt[3]{90}}{6} \\ \left(c\right) \frac{5\sqrt[3]{81y^2}}{9y} \end{gathered}$$

Part (a) Step 1. Given information.

Given expression:

$\frac{1}{\sqrt[3]{7}}$

Part (a) Step 2. Finding solution.

Finding the solution.

Multiply numerator and denominator by the same factor.

Multiply numerator and denominator $\sqrt[3]{7^2}$.

$$\begin{gathered} \frac{1}{\sqrt[3]{7}}=\frac{1}{\sqrt[3]{7}}\times \frac{\sqrt[3]{7^2}}{\sqrt[3]{7^2}} \\ =\frac{\sqrt[3]{7^2}}{\sqrt[3]{7^3}} \\ =\frac{\sqrt[3]{7^2}}{7} \\ =\frac{\sqrt[3]{49}}{7} \end{gathered}$$

Part (b) Step 1. Given information.

Given expression:

$\sqrt[3]{\frac{5}{12}}$

Part (b) Step 2. Finding the solution.

Finding the solution.

Multiply numerator and denominator by the same factor.

Multiply numerator and denominator $\sqrt[3]{{12}^2}$.

$$\begin{gathered} \frac{\sqrt[3]{5}}{\sqrt[3]{12}}=\frac{\sqrt[3]{5}}{\sqrt[3]{12}}\times \frac{\sqrt[3]{{12}^2}}{\sqrt[3]{{12}^2}} \\ =\frac{\sqrt[3]{5.{12}^2}}{12} \\ =\frac{2\sqrt[3]{90}}{6} \end{gathered}$$

Part (c) Step 1. Given information.

Given expression :

$\frac{5\sqrt[3]{81y^2}}{9y}$

Part (c) Step 2. Finding the solution.

Finding the solution.

Multiply numerator and denominator by the same factor.

Multiply numerator and denominator $\sqrt[3]{9^2{y}^2}$.

$$\begin{gathered} \frac{5}{\sqrt[3]{9y}}=\frac{5}{\sqrt[3]{9y}}\times \frac{\sqrt[3]{9^2{y}^2}}{\sqrt[3]{9^2{y}^2}} \\ =\frac{5\sqrt[3]{9^2{y}^2}}{9y} \\ =\frac{5\sqrt[3]{81y^2}}{9y} \end{gathered}$$