The value of \(\frac{2 \sqrt{2}}{2-\sqrt{2}}\) is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(2(\sqrt{2}+1)\)
A surd in the denominator is removed by multiplying the top and bottom by its conjugate, here \(2+\sqrt{2}\), because \((a-b)(a+b)=a^2-b^2\) has no root left in it. Multiplying by this fraction does not change the value, since it equals 1.
\(\displaystyle \frac{2\sqrt{2}}{2-\sqrt{2}}\times\frac{2+\sqrt{2}}{2+\sqrt{2}}=\frac{2\sqrt{2}(2+\sqrt{2})}{4-2}\)
The denominator is now a whole number: \(\displaystyle \frac{4\sqrt{2}+4}{2}=2\sqrt{2}+2=2(\sqrt{2}+1)\)