Question

Simplify the expression
\[\frac{1}{\sqrt{36} - \sqrt{27}} - \frac{1}{\sqrt{27} - \sqrt{18}} + \frac{1}{\sqrt{18} - \sqrt{9}}.\]

Answer 1

Compute the square roots: √36 = 6, √27 = 3√3, √18 = 3√2, √9 = 3. Factor 3 from each denominator:
\[
\frac{1}{\sqrt{36}-\sqrt{27}}-\frac{1}{\sqrt{27}-\sqrt{18}}+\frac{1}{\sqrt{18}-\sqrt{9}}
=\frac{1}{3}\Big(\frac{1}{2-\sqrt3}-\frac{1}{\sqrt3-\sqrt2}+\frac{1}{\sqrt2-1}\Big).
\]
Rationalize each term: 1/(2−√3)=2+√3, 1/(√3−√2)=√3+√2, 1/(√2−1)=√2+1. Thus the bracket equals
(2+√3) − (√3+√2) + (√2+1) = 3,
and the whole expression is (1/3)·3 = 1.