Express {8 - 3{6}}{2{3} + 3{2}} in the form p{3} + q{2}.

Correct Answer is : 72−17337{2} - {17{3}}{3}72−3173

Correct Answer is : 72−17337{2} - {17{3}}{3}72−3173

Test Index

WAEC Further Mathematics 2016 Paper

Question : 2

Total: 40

Express

\frac{8 - 3\sqrt{6}}{2\sqrt{3} + 3\sqrt{2}}

p\sqrt{3} + q\sqrt{2}

Solution:

Concept:

Rationalising the denominator by multiplying with its conjugate simplifies the expression into the required form

p\sqrt{3} + q\sqrt{2}

Explanation:

The conjugate of

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

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

Multiply the numerator and denominator by this conjugate:

\frac{8 - 3\sqrt{6}}{2\sqrt{3} + 3\sqrt{2}} \times \frac{2\sqrt{3} - 3\sqrt{2}}{2\sqrt{3} - 3\sqrt{2}}

Expand the numerator:

(8 - 3\sqrt{6})(2\sqrt{3} - 3\sqrt{2})

= 16\sqrt{3} - 24\sqrt{2} - 18\sqrt{2} + 18\sqrt{3}

= 34\sqrt{3} - 42\sqrt{2}

Simplify the denominator using the difference of squares:

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

Therefore:

\frac{34\sqrt{3} - 42\sqrt{2}}{-6} = -\frac{17}{3}\sqrt{3} + 7\sqrt{2}

= 7\sqrt{2} - \frac{17\sqrt{3}}{3}

Answer:

7\sqrt{2} - \frac{17\sqrt{3}}{3}

This matches option B.

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