Simplify (4 – frac{1}{2 – sqrt{3}})
-
A.
2(sqrt{3}) -
B.
-2 – (sqrt{3}) -
C.
-2 + (sqrt{3}) -
D.
2 – (sqrt{3})
Correct Answer: Option D
Explanation
(4 – frac{1}{2 – sqrt{3}} = frac{4(2 – sqrt{3}) – 1}{2 – sqrt{3}})
= (frac{8 – 4sqrt{3} – 1}{2 – sqrt{3}})
= (frac{7 – 4sqrt{3}}{2 – sqrt{3}})
Rationalizing,
((frac{7 – 4sqrt{3}}{2 – sqrt{3}}) (frac{2 + sqrt{3}}{2 + sqrt{3}}))
= (frac{14 + 7sqrt{3} – 8sqrt{3} – 12}{4 + 2sqrt{3} – 2sqrt{3} – 3})
= (frac{2 – sqrt{3}}{1} = 2 – sqrt{3})