Question

$$\text { Simplify } \frac{3}{5-\sqrt{5}}$$ \(\begin{array}{llll}\text { (A) } \frac{15+3 \sqrt{5}}{20} & \text { (B) } \frac{15+\sqrt{5}}{20} & \text { (C) } \frac{15+\sqrt{15}}{20}\end{array}\) (D) \(\frac{15-3 \sqrt{5}}{20}\)

Short answer

(A) \(\frac{15+3 \sqrt{5}}{20}\)

Step-by-step solution

Multiply by the Conjugate

To deal with the radical in the denominator, the expression is multiplied by the conjugate of the denominator over itself, i.e. \((5+\sqrt{5})/(5+\sqrt{5})\).

Simplify the Numerator

The numerator becomes \(3*(5+\sqrt{5}) = 15+3\sqrt{5}\).

Simplify the Denominator

The denominator becomes \((5-\sqrt{5})*(5+\sqrt{5}) = 5^2 - (\sqrt{5})^2 = 25 - 5 = 20\).

Write the Simplified Fraction

The simplified fraction is \((15+3\sqrt{5})/20\).