Q. 259 - Intermediate Algebra

Question

In the following exercises, rationalize the denominator.

$$\begin{gathered} a) \frac{10}{\sqrt{6}} \\ b) \sqrt{\frac{4}{27}} \\ c) \frac{10}{\sqrt{5x}} \end{gathered}$$

Step-by-Step Solution

Answer

Part(a) The rationalization of the denominator is $\frac{5\sqrt{6}}{3}$.

Part(b) The rationalization of the denominator is $\frac{2\sqrt{3}}{9}$.

Part(c) The rationalization of the denominator is $\frac{2\sqrt{5x}}{x}$.

Step 1. Given information.

The Given expressions are,

$$\begin{gathered} a) \frac{10}{\sqrt{6}} \\ b) \sqrt{\frac{4}{27}} \\ c) \frac{10}{\sqrt{5x}} \end{gathered}$$

Step 2. Solving part(a).

Now, to rationalize the expression first we will multiply numerator and denominator by $\sqrt{6}$.

$\frac{10}{\sqrt{6}}=\frac{10\times \sqrt{6}}{\sqrt{6}\times \sqrt{6}}$

On simplifying we get,

$\frac{10\times \sqrt{6}}{\sqrt{6}\times \sqrt{6}}=\frac{10\sqrt{6}}{6}$

Then removing the common factors.

$\frac{10\sqrt{6}}{6}=\frac{2\sqrt{6}}{3}$

Hence, the rationalization of the denominator is $\frac{2\sqrt{6}}{3}$.

Step 3. Solving part(b).

Now, to rationalize the expression first we will rewrite the expression using quotient property and simplify.

$$\begin{gathered} \sqrt{\frac{4}{27}}=\frac{\sqrt{4}}{\sqrt{27}} \\ =\frac{2}{3\sqrt{3}} \end{gathered}$$

Multiplying numerator and denominator by $\sqrt{3}$.

$\frac{2}{3\sqrt{3}}=\frac{2\times \sqrt{3}}{3\sqrt{3}\times \sqrt{3}}$

On simplifying we get,

$\frac{2\times \sqrt{3}}{3\sqrt{3}\times \sqrt{3}}=\frac{2\sqrt{3}}{9}$

Hence, the rationalization of the denominator is $\frac{2\sqrt{3}}{9}$.

Step 4. Solving part(c).

Now, to rationalize the expression first we will multiply numerator and denominator by $\sqrt{5x}$.

$\frac{10}{\sqrt{5x}}=\frac{10\times \sqrt{5x}}{\sqrt{5x}\times \sqrt{5x}}$

On simplifying we get,

$\frac{10\times \sqrt{5x}}{\sqrt{5x}\times \sqrt{5x}}=\frac{10\sqrt{5x}}{5x}$

Then removing the common factors.

$\frac{10\sqrt{5x}}{5x}=\frac{2\sqrt{5x}}{x}$

Hence, the rationalization of the denominator is $\frac{2\sqrt{5x}}{x}$.