What is the simplified form of the expression \frac{12}{\sqrt{2} + \sqrt{3} - \sqrt{5}}? — JAMB UTME

What is the simplified form of the expression $\frac{12}{\sqrt{2} + \sqrt{3} - \sqrt{5}}$ ?

3\sqrt{2} + 2\sqrt{3} + \sqrt{30}

By grouping the denominator as

(\sqrt{2}+\sqrt{3}) - \sqrt{5}

and multiplying by its conjugate

(\sqrt{2}+\sqrt{3}) + \sqrt{5}

, the denominator reduces to

2\sqrt{6}

. Multiplying the resulting fraction by

\sqrt{6}/\sqrt{6}

yields

\sqrt{12} + \sqrt{18} + \sqrt{30}

, which simplifies directly to

3\sqrt{2} + 2\sqrt{3} + \sqrt{30}

Group the terms in the denominator as

((\sqrt{2} + \sqrt{3}) - \sqrt{5})

and multiply the numerator and denominator by its conjugate

((\sqrt{2} + \sqrt{3}) + \sqrt{5})

The fraction becomes

\frac{12((\sqrt{2} + \sqrt{3}) + \sqrt{5})}{((\sqrt{2} + \sqrt{3}) - \sqrt{5})((\sqrt{2} + \sqrt{3}) + \sqrt{5})}

Applying the difference of two squares to eliminate the outer radical.

Expand the denominator using

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

Denominator =

(\sqrt{2} + \sqrt{3})^2 - (\sqrt{5})^2 = (2 + 2\sqrt{6} + 3) - 5 = 2\sqrt{6}

Simplifying the algebraic square of a binomial surd.

Divide the numerator by the constant factor of the denominator.

\frac{12(\sqrt{2} + \sqrt{3} + \sqrt{5})}{2\sqrt{6}} = \frac{6(\sqrt{2} + \sqrt{3} + \sqrt{5})}{\sqrt{6}}.

Simplifying numerical coefficients before further rationalization.

Rationalize the remaining monomial radical in the denominator by multiplying numerator and denominator by

\sqrt{6}

\frac{6(\sqrt{12} + \sqrt{18} + \sqrt{30})}{6} = \sqrt{12} + \sqrt{18} + \sqrt{30}.

Eliminating

\sqrt{6}

from the denominator.

Simplify each radical to its simplest surd form.

\sqrt{12} = 2\sqrt{3}, \quad \sqrt{18} = 3\sqrt{2}, \quad \text{so } \sqrt{12} + \sqrt{18} + \sqrt{30} = 3\sqrt{2} + 2\sqrt{3} + \sqrt{30}.

Factoring out perfect square components from radical terms.

Rationalisation of trinomial surd denominators using repeated conjugate multiplication

Estimated Time:2m 0s

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