Question

rationalize the denominator $\frac{1}{3-sqrt{5}}$.
$\frac{1}{3-sqrt{5}} = square$

Explanation:

Step1: Multiply by conjugate

Multiply numerator and denominator by $3+\sqrt{5}$, the conjugate of $3-\sqrt{5}$.
$$\frac{1}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}} = \frac{3+\sqrt{5}}{(3-\sqrt{5})(3+\sqrt{5})}$$

Step2: Expand denominator

Use difference of squares: $(a-b)(a+b)=a^2-b^2$.
$$\frac{3+\sqrt{5}}{3^2 - (\sqrt{5})^2} = \frac{3+\sqrt{5}}{9 - 5}$$

Step3: Simplify denominator

Calculate the value of the denominator.
$$\frac{3+\sqrt{5}}{4}$$

Answer:

$\frac{3+\sqrt{5}}{4}$