Find the value of 3 + 1/√3 + 1/(√3 + 3) + 1/(√3 – 3). (a) 6 (b) 3 (c) 3/(2(√3 + 3)) (d) 2√3

Solution

To find the value of the given expression

\[
3+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{3}+3}+\frac{1}{\sqrt{3}-3},
\]

we can simplify each term individually, starting with rationalizing the denominators where necessary:

Simplification Steps

First, let’s address the fractions involving square roots by multiplying the numerator and denominator by the conjugate of the denominator when needed:

\[
\begin{aligned}
& 3+\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}+\frac{1}{3+\sqrt{3}} \times \frac{3-\sqrt{3}}{3-\sqrt{3}}+\frac{1}{\sqrt{3}-3} \times \frac{\sqrt{3}+3}{\sqrt{3}+3} \\
& = 3+\frac{\sqrt{3}}{3}+\frac{3-\sqrt{3}}{3^2-(\sqrt{3})^2}+\frac{\sqrt{3}+3}{(\sqrt{3})^2-3^2} \\
& = 3+\frac{\sqrt{3}}{3}+\frac{3-\sqrt{3}}{9-3}+\frac{\sqrt{3}+3}{3-9} \\
& = 3+\frac{\sqrt{3}}{3}+\frac{3-\sqrt{3}}{6}-\frac{\sqrt{3}+3}{6} \\
\end{aligned}
\]

Combining terms:

\[
\begin{aligned}
& = \frac{3 \times 6}{6}+\frac{2 \sqrt{3}}{6}+\frac{3-\sqrt{3}-\sqrt{3}-3}{6} \\
& = \frac{18+2 \sqrt{3}-2 \sqrt{3}}{6} \\
& = \frac{18}{6} \\
& = 3 \\
\end{aligned}
\]

Therefore, the value of the given expression is 3.

The correct answer is (b) 3.