$\frac{1}{√1+√2}$+$\frac{1} {√2 + √3}$+ $\frac{1}{√3 + √4}$

asked by guest
on Nov 16, 2024 at 12:17 am



You asked:

Evaluate the expression: \(\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}}\)

MathBot Answer:

\[\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} = \frac{1}{\sqrt{3} + 2} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{1 + \sqrt{2}} \approx 1.0\]

\[\begin{aligned}\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}}& = \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}}\\& = \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + 2}\\& = \frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{2 + \sqrt{3}}\\& = \left(\frac{1}{1 + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}}\right) + \frac{1}{2 + \sqrt{3}}\\& = \frac{1}{1 + \sqrt{2}} + \frac{1}{2 + \sqrt{3}} + \frac{1}{\sqrt{2} + \sqrt{3}}\end{aligned}\]

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