If \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} - \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} +… — JAMB UTME

If $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} - \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} = x\sqrt{15}$ , what is the value of $x$ ?

Answer: 2

The value of

x

is 2.

Rationalising both fractions yields

4 + \sqrt{15}

and

4 - \sqrt{15}

. Subtracting the second from the first gives

(4 + \sqrt{15}) - (4 - \sqrt{15}) = 2\sqrt{15}

. Comparing

2\sqrt{15}

with

x\sqrt{15}

gives

x = 2

Rationalise the denominator of the first fraction

\frac{(\sqrt{5}+\sqrt{3})^2}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})} = \frac{5 + 2\sqrt{15} + 3}{5 - 3} = 4 + \sqrt{15}

Multiplying numerator and denominator by the conjugate of the denominator removes the surd from the denominator.

Rationalise the denominator of the second fraction

\frac{(\sqrt{5}-\sqrt{3})^2}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} = \frac{5 - 2\sqrt{15} + 3}{5 - 3} = 4 - \sqrt{15}

Multiply by the conjugate

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

to simplify the second surd term.

Subtract the simplified expressions

(4 + \sqrt{15}) - (4 - \sqrt{15}) = 4 - 4 + \sqrt{15} + \sqrt{15} = 2\sqrt{15}

Distribute the negative sign and combine like surd terms.

Solve for the unknown coefficient x

2\sqrt{15} = x\sqrt{15} \implies x = 2

Divide both sides of the equation by

\sqrt{15}

to isolate

x

Binomial Surd Rationalisation and Simplification

Estimated Time:1m 30s

Question