0606 P22 - Mar 2023 - Q2 - 4 marks
7771
Do not use a calculator in this question.
Expand and simplify
\(\left(\frac{x\sqrt{11}}{2\sqrt3-1}\right)^2,\)
giving your answer with a rational denominator.
Solution
Answer: \(\displaystyle \frac{x^2(13+4\sqrt3)}{11}\).
Answer: \(\displaystyle \frac{x^2(13+4\sqrt3)}{11}\).
First square the numerator and denominator:
\(\left(\frac{x\sqrt{11}}{2\sqrt3-1}\right)^2 =\frac{11x^2}{(2\sqrt3-1)^2}.\)
Now
\((2\sqrt3-1)^2=12-4\sqrt3+1=13-4\sqrt3.\)
So
\(\left(\frac{x\sqrt{11}}{2\sqrt3-1}\right)^2 =\frac{11x^2}{13-4\sqrt3}.\)
Rationalise the denominator by multiplying by \(13+4\sqrt3\):
\(\frac{11x^2}{13-4\sqrt3}\cdot \frac{13+4\sqrt3}{13+4\sqrt3} =\frac{11x^2(13+4\sqrt3)}{13^2-(4\sqrt3)^2}.\)
The denominator is
\(169-48=121.\)
Hence
\(\frac{11x^2(13+4\sqrt3)}{121} =\frac{x^2(13+4\sqrt3)}{11}.\)
