What is the simplified form of the surd expression \frac{3\sqrt{2} + 2\sqrt{3}}{3\sqrt{2} … — JAMB UTME

What is the simplified form of the surd expression $\frac{3\sqrt{2} + 2\sqrt{3}}{3\sqrt{2} - 2\sqrt{3}}$ after rationalizing the denominator?

5 + 2\sqrt{6}

To rationalize

\frac{3\sqrt{2} + 2\sqrt{3}}{3\sqrt{2} - 2\sqrt{3}}

, we multiply both the numerator and denominator by the conjugate of the denominator,

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

. Expanding the numerator yields

(3\sqrt{2})^2 + 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 18 + 12\sqrt{6} + 12 = 30 + 12\sqrt{6}

. Expanding the denominator gives

(3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6

. Dividing

30 + 12\sqrt{6}

6

gives

5 + 2\sqrt{6}

Identify the conjugate of the denominator

The conjugate of

3\sqrt{2} - 2\sqrt{3}

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

Multiplying by the conjugate converts the binomial denominator into a rational number using the difference of squares identity

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

Multiply the numerator and denominator by the conjugate

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

This maintains expression equivalence while eliminating radicals from the denominator.

Expand the numerator and denominator independently

Numerator:

(3\sqrt{2})^2 + 2(3\sqrt{2})(2\sqrt{3}) + (2\sqrt{3})^2 = 18 + 12\sqrt{6} + 12 = 30 + 12\sqrt{6}

. Denominator:

(3\sqrt{2})^2 - (2\sqrt{3})^2 = 18 - 12 = 6

Apply algebraic expansion

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

and simplification of radicals.

Divide each term in the numerator by the denominator

\frac{30 + 12\sqrt{6}}{6} = \frac{30}{6} + \frac{12\sqrt{6}}{6} = 5 + 2\sqrt{6}.

Simplify the fraction to express the answer in standard surd form

a + b\sqrt{c}

Rationalization of Binomial Denominators

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