Simplify: ≤ft({2+{3}}{2-{3}}+{2-{3}}{2+{3}}+{{3}-1}{{3}+1})

Correct Answer is : 16−316 - {3}16−3

Correct Answer is : 16−316 - {3}16−3

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Powers and Roots Practice Test 5

Question : 8 of 31

Marks: +1, -0

Simplify:

\left(\frac{2+\sqrt{3}}{2-\sqrt{3}}+\frac{2-\sqrt{3}}{2+\sqrt{3}}+\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)

Solution:

Concept:

Simplify the expression by rationalizing the denominators and combining like terms.

Explanation:

Combine the first two fractions using a common denominator.

\frac{2+\sqrt{3}}{2-\sqrt{3}} + \frac{2-\sqrt{3}}{2+\sqrt{3}} = \frac{(2+\sqrt{3})^2+(2-\sqrt{3})^2}{(2-\sqrt{3})(2+\sqrt{3})}

Simplify the numerator and denominator.

(2+\sqrt{3})^2+(2-\sqrt{3})^2 = (7+4\sqrt{3})+(7-4\sqrt{3})=14

(2-\sqrt{3})(2+\sqrt{3})=4-3=1

So the first two terms sum to

14

Now simplify the third term by rationalizing.

\frac{\sqrt{3}-1}{\sqrt{3}+1} = \frac{(\sqrt{3}-1)^2}{(\sqrt{3}+1)(\sqrt{3}-1)} = \frac{4-2\sqrt{3}}{2} = 2-\sqrt{3}

Add all parts.

14 + (2-\sqrt{3}) = 16-\sqrt{3}

Answer:

16-\sqrt{3}

, which matches option D.

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