Question

Rationalize the denominator.

$\frac{1}{5-\sqrt{3}}$

Short answer

The solution is$\frac{5+\sqrt{3}}{22}$

Step-by-step solution

Step 1. Apply the concept of rationalizing the denominator.

If a fraction has a denominator of the form $A+B\sqrt{C}$, we canrationalize the denominator by multiplying numerator and denominator by the conjugate radical $A-B\sqrt{C}$.

Step 2. Apply the concept of special multiplication.

If A and B are real numbers, then

$\left(A-B\right)\left(A+B\right)=A^2-B^2$

Step 3. Analyze the expression.

Given

$\frac{1}{5-\sqrt{3}}$

Here the denominator is $5-\sqrt{3}$, so multiply the numerator and denominator by$5+\sqrt{3}$

Step 4. Rationalize the denominator.

Consider$\frac{1}{5-\sqrt{3}}$

So, the simplified expression after rationalizing denominator is given by:

role="math" localid="1643224442536" $\begin{array}{l}\frac{1}{5-\sqrt{3}}=\frac{1}{5-\sqrt{3}}\cdot \frac{5+\sqrt{3}}{5+\sqrt{3}}\\ \Rightarrow \frac{1}{5-\sqrt{3}}=\frac{5+\sqrt{3}}{\left(5-\sqrt{3}\right)\left(5+\sqrt{3}\right)}\\ \Rightarrow \frac{1}{5-\sqrt{3}}=\frac{5+\sqrt{3}}{5^2-{\left(\sqrt{3}\right)}^2}\\ \Rightarrow \frac{1}{5-\sqrt{3}}=\frac{5+\sqrt{3}}{25-3}\\ \Rightarrow \frac{1}{5-\sqrt{3}}=\frac{5+\sqrt{3}}{22}\end{array}$