Question

Rationalize the numerator.

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

Short answer

The solution is$\frac{-4}{3\left(1+\sqrt{5}\right)}$

Step-by-step solution

Step 1. Apply the concept of rationalizing the numerator.

If a fraction has a numerator of the form$A+B\sqrt{C}$, we can rationalize the numerator 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

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

Here the numerator is $1-\sqrt{5}$, so multiply the numerator and denominator by$1+\sqrt{5}$

Step 4. Rationalize the numerator.

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

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

role="math" localid="1643278790354" $\begin{array}{l}\frac{1-\sqrt{5}}{3}=\frac{1-\sqrt{5}}{3}\cdot \frac{1+\sqrt{5}}{1+\sqrt{5}}\\ \Rightarrow \frac{1-\sqrt{5}}{3}=\frac{\left(1-\sqrt{5}\right)\left(1+\sqrt{5}\right)}{3\left(1+\sqrt{5}\right)}\\ \Rightarrow \frac{1-\sqrt{5}}{3}=\frac{1^2-{\left(\sqrt{5}\right)}^2}{3\left(1+\sqrt{5}\right)}\\ \Rightarrow \frac{1-\sqrt{5}}{3}=\frac{1-5}{3\left(1+\sqrt{5}\right)}\\ \Rightarrow \frac{1-\sqrt{5}}{3}=\frac{-4}{3\left(1+\sqrt{5}\right)}\end{array}$