Question : The value of $5–\frac{8+2\sqrt{15}}{4}–\frac{1}{8+2\sqrt{15}}$ is equal to:
Option 1: $\frac{1}{4}$
Option 2: $1$
Option 3: $\frac{2}{3}$
Option 4: $\frac{1}{2}$
Correct Answer: $1$
Solution :
Given: The expression is $5–\frac{8+2\sqrt{15}}{4}–\frac{1}{8+2\sqrt{15}}$.
Rationalize the fraction,
$\frac{1}{8+2\sqrt{15}}=\frac{1\times (8–2\sqrt{15})}{(8+2\sqrt{15})\times (8–2\sqrt{15})}$
= $\frac{8–2\sqrt{15}}{64–60}$
= $\frac{8–2\sqrt{15}}{4}$
The expression is $5–\frac{8+2\sqrt{15}}{4}–\frac{8–2\sqrt{15}}{4}$.
= $\frac{20–8–2\sqrt{15}–8+2\sqrt{15}}{4}$
= $\frac{20–16}{4}$
= $\frac{20–16}{4}$
= $\frac{4}{4}$
= 1
Hence, the correct answer is $1$.
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