Question : If $\frac{8+2 \sqrt{3}}{3 \sqrt{3}+5}=a \sqrt{3}–b$, then the value of $a + b$ is equal to:
Option 1: 18
Option 2: 15
Option 3: 24
Option 4: 16
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Correct Answer: 18
Solution : Given: $\frac{8+2 \sqrt{3}}{3 \sqrt{3}+5}=a \sqrt{3}–b$ Rationalize the given fraction, $\frac{(8+2 \sqrt{3})\times(3\sqrt3–5)}{(3 \sqrt{3}+5)\times (3\sqrt3–5)}=\frac{24\sqrt3–40+18–10\sqrt3}{27–25}$ ⇒ $\frac{14\sqrt3–22}{2}=7\sqrt3–11$ Now, compare the value with the given expression $a \sqrt{3}–b$, we get, $7\sqrt3–11=a \sqrt{3}–b$ ⇒ $a=7,b=11$ The value of $a + b=7+11=18$. Hence, the correct answer is 18.
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