Q. 261 - Intermediate Algebra

Question

In the following exercises, rationalize the denominator.  

$$\begin{gathered} a) \frac{6}{\sqrt{7}} \\ b) \sqrt{\frac{8}{45}} \\ c)\frac{12}{\sqrt{3p}} \end{gathered}$$

Step-by-Step Solution

Answer

Part(a) The rationalization of the denominator is $\frac{6\sqrt{7}}{7}$.

Part(b) The rationalization of the denominator is $\frac{2\sqrt{10}}{15}$.

Part(c) The rationalization of the denominator is $\frac{4\sqrt{3p}}{p}$.

Step 1. Given information.

The given expressions are,

$$\begin{gathered} a) \frac{6}{\sqrt{7}} \\ b) \sqrt{\frac{8}{45}} \\ c)\frac{12}{\sqrt{3p}} \end{gathered}$$ 

Step 2. Solving part (a).

Now, to rationalize the expression first we will multiply the numerator and denominator by $\sqrt{7}$.

$\frac{6}{\sqrt{7}}=\frac{6\times \sqrt{7}}{\sqrt{7}\times \sqrt{7}}$

On simplifying we get,

$\frac{6\times \sqrt{7}}{\sqrt{7}\times \sqrt{7}}=\frac{6\sqrt{7}}{7}$

Hence, the rationalization of the denominator is $\frac{6\sqrt{7}}{7}$.

Step 3. Solving part (b).

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

$$\begin{gathered} \sqrt{\frac{8}{45}}=\frac{\sqrt{8}}{\sqrt{45}} \\ =\frac{2\sqrt{2}}{3\sqrt{5}} \end{gathered}$$

Multiplying the numerator and denominator by $\sqrt{5}$.

$\frac{2\sqrt{2}}{3\sqrt{5}}=\frac{2\sqrt{2}\times \sqrt{5}}{3\sqrt{5}\times \sqrt{5}}$

Then on simplifying we get 

$\frac{2\sqrt{2}\times \sqrt{5}}{3\sqrt{5}\times \sqrt{5}}=\frac{2\sqrt{10}}{15}$

Hence, the rationalization of the denominator is $\frac{2\sqrt{10}}{15}$.

Step 4. Solving part (c).

Now, to rationalize the expression first we will multiply the numerator and denominator by $\sqrt{3p}$.

$\frac{12}{\sqrt{3p}}=\frac{12\times \sqrt{3p}}{\sqrt{3p}\times \sqrt{3p}}$

On simplifying we get,

$\frac{12\times \sqrt{3p}}{\sqrt{3p}\times \sqrt{3p}}=\frac{12\sqrt{3p}}{3p}$

Then removing the common factors.

$\frac{12\sqrt{3p}}{3p}=\frac{4\sqrt{3p}}{p}$

Hence, the rationalization of the denominator is $\frac{4\sqrt{3p}}{p}$.