Question
Rationalize the denominator : $\frac{1}{3 \sqrt{5}+2 \sqrt{2}}$
✓
Answer
$\frac{1}{3 \sqrt{5}+2 \sqrt{2}}=\frac{1}{(3 \sqrt{5}+2 \sqrt{2})} \times \frac{(3 \sqrt{5}-2 \sqrt{2})}{(3 \sqrt{5}-2 \sqrt{2})}$
...[Multiplying the numerator and denominator by $(3 \sqrt{5}-2 \sqrt{2})$ ]
$
\begin{aligned}
& =\frac{1 \times(3 \sqrt{5}-2 \sqrt{2})}{(3 \sqrt{5}+2 \sqrt{2})(3 \sqrt{5}-2 \sqrt{2})} \\
& =\frac{3 \sqrt{5}-2 \sqrt{2}}{(3 \sqrt{5})^2-(2 \sqrt{2})^2} \\
& =\frac{3 \sqrt{5}-2 \sqrt{2}}{(9 \times 5)-(4 \times 2)} \\
& =\frac{3 \sqrt{5}-2 \sqrt{2}}{45-8} \\
\therefore \quad \frac{1}{3 \sqrt{5}+2 \sqrt{2}} & =\frac{3 \sqrt{5}-2 \sqrt{2}}{37}
\end{aligned}
$
...[Multiplying the numerator and denominator by $(3 \sqrt{5}-2 \sqrt{2})$ ]
$
\begin{aligned}
& =\frac{1 \times(3 \sqrt{5}-2 \sqrt{2})}{(3 \sqrt{5}+2 \sqrt{2})(3 \sqrt{5}-2 \sqrt{2})} \\
& =\frac{3 \sqrt{5}-2 \sqrt{2}}{(3 \sqrt{5})^2-(2 \sqrt{2})^2} \\
& =\frac{3 \sqrt{5}-2 \sqrt{2}}{(9 \times 5)-(4 \times 2)} \\
& =\frac{3 \sqrt{5}-2 \sqrt{2}}{45-8} \\
\therefore \quad \frac{1}{3 \sqrt{5}+2 \sqrt{2}} & =\frac{3 \sqrt{5}-2 \sqrt{2}}{37}
\end{aligned}
$
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