Rationalising a surd denominator

Number & surds

(1MA1) · Higher

ORIGINAL

Multiply by the conjugate to clear a surd from the bottom of a fraction.

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Question

Rationalise the denominator and simplify

\dfrac{5 + \sqrt{3}}{2 - \sqrt{3}}

Every step worked, with the reasoning.

1
$\dfrac{5 + \sqrt{3}}{2 - \sqrt{3}} \times \dfrac{2 + \sqrt{3}}{2 + \sqrt{3}}$
Multiply top and bottom by the conjugate of the denominator, $2 + \sqrt{3}$ .
2
Numerator: $(5 + \sqrt{3})(2 + \sqrt{3}) = 10 + 5\sqrt{3} + 2\sqrt{3} + 3 = 13 + 7\sqrt{3}$
Expand the numerator and collect the surd terms.
3
Denominator: $(2 - \sqrt{3})(2 + \sqrt{3}) = 4 - 3 = 1$
The conjugate pair is a difference of two squares, so the surd cancels.
4
$\dfrac{13 + 7\sqrt{3}}{1} = 13 + 7\sqrt{3}$
Divide by 1.

Answer: $13 + 7\sqrt{3}$