Q. 8.103 - Intermediate Algebra

Question

In the following exercise ,simplify 

$$\begin{gathered} \left(a\right) \frac{1}{\sqrt[4]{3}} \\ \left(b\right) \sqrt[4]{\frac{3}{64}} \\ \left(c\right) \frac{3}{\sqrt[4]{125x}} \end{gathered}$$

Step-by-Step Solution

Answer

$$\begin{gathered} \left(a\right) \frac{\sqrt[4]{27}}{3} \\ \left(b\right) \frac{\sqrt[4]{12}}{4} \\ \left(c\right) \frac{3\sqrt[4]{5x^3}}{5x} \end{gathered}$$ 

Part (a) Step 1. Given information.

Given expression:

$\frac{1}{\sqrt[4]{3}}$

Part (a) Step 2. Finding solution.

Finding the solution.

Multiply numerator and denominator by the same factor.

Multiply numerator and denominator $\sqrt[4]{3^3}$.

$$\begin{gathered} \frac{1}{\sqrt[4]{3}}\times \frac{\sqrt[4]{3^3}}{\sqrt[4]{3^3}} \\ =\frac{\sqrt[4]{27}}{3} \end{gathered}$$

Part (b) Step 1. Given information.

Given expression:

$\sqrt[4]{\frac{3}{64}}$

Part (b) Step 2. Finding the solution.

Finding the solution.

Multiply numerator and denominator by the same factor.

Multiply numerator and denominator $\sqrt[4]{4^3}$.$$\begin{gathered} \frac{\sqrt[4]{3}}{\sqrt[4]{64}}=\frac{\sqrt[4]{3}}{\sqrt[4]{2^4.2^2}} \\ =\frac{\sqrt[4]{3}}{2.\sqrt[4]{4}}\times \frac{\sqrt[4]{4^3}}{\sqrt[4]{4^3}} \\ =\frac{\sqrt[4]{3}}{2.4}.\sqrt[4]{4^3} \\ =\frac{\sqrt[4]{3}}{8}.\sqrt[4]{2^4.2^2} \\ =\frac{\sqrt[4]{3}}{4}.\sqrt[4]{4} \end{gathered}$$

Part (c) Step 1. Given information.

Given expression :

$\frac{3}{\sqrt[4]{125x}}$

Part (c) Step 2. Finding the solution.

Finding the solution.

Multiply numerator and denominator by the same factor.

Multiply numerator and denominator $\frac{\sqrt[4]{5x^3}}{\sqrt[4]{5x^3}}$.

$$\begin{gathered} \frac{3}{\sqrt[4]{125x}}=\frac{3}{\sqrt[4]{5^{3}x}} \\ =\frac{3}{\sqrt[4]{5^{3}x}}\times \frac{\sqrt[4]{5x^3}}{\sqrt[4]{5x^3}} \\ =\frac{3\sqrt[4]{5x^3}}{5x} \end{gathered}$$