Simplify the surd expression \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}. — JAMB UTME

Simplify the surd expression $\frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}}$ .

\frac{7 + 2\sqrt{10}}{3}

Multiplying both the numerator and denominator by the conjugate of the denominator,

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

, expands the numerator to

5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10}

and simplifies the denominator using the difference of squares to

5 - 2 = 3

, giving

\frac{7 + 2\sqrt{10}}{3}

Identify the conjugate of the denominator

The conjugate of

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

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

Rationalizing a binomial denominator requires multiplying by its conjugate to apply the difference of squares identity

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

Multiply both numerator and denominator by the conjugate

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

This maintains the value of the fraction while removing radical terms from the denominator.

Expand the numerator and denominator

Numerator:

(\sqrt{5})^2 + 2\sqrt{5}\sqrt{2} + (\sqrt{2})^2 = 5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10}

. Denominator:

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

Apply algebraic expansion

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

and difference of squares.

Combine terms to form the final simplified expression

\frac{7 + 2\sqrt{10}}{3}

The expression is now fully rationalized and in standard simplified surd form.

Rationalization of Binomial Denominators

Question