What is the simplified form of the expression \frac{\sqrt{5}}{\sqrt{5} - \sqrt{3}}? — JAMB UTME

What is the simplified form of the expression $\frac{\sqrt{5}}{\sqrt{5} - \sqrt{3}}$ ?

\frac{5 + \sqrt{15}}{2}

To rationalize the denominator of

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

, multiply both numerator and denominator by the conjugate

\sqrt{5} + \sqrt{3}

. The numerator becomes

\sqrt{5}(\sqrt{5} + \sqrt{3}) = 5 + \sqrt{15}

and the denominator becomes

(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2

, yielding

\frac{5 + \sqrt{15}}{2}

Identify the conjugate of the denominator

The conjugate of

\sqrt{5} - \sqrt{3}

\sqrt{5} + \sqrt{3}

Multiplying a binomial surd by its conjugate eliminates the radical terms in the denominator using the difference of two squares identity

(a-b)(a+b) = a^2 - b^2

Multiply both the numerator and the denominator by the conjugate

\frac{\sqrt{5}(\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3})(\sqrt{5} + \sqrt{3})}

Multiplying the fraction by

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

is equivalent to multiplying by

1

, preserving the value of the expression.

Expand the numerator and simplify the denominator

Numerator:

\sqrt{5} \times \sqrt{5} + \sqrt{5} \times \sqrt{3} = 5 + \sqrt{15}

. Denominator:

(\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2

Applying the distributive law to the numerator and difference of squares to the denominator simplifies both parts.

Combine the simplified numerator and denominator

\frac{5 + \sqrt{15}}{2}

This is the simplified rationalized form.

Rationalization of Denominators

Question