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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