Simplify the expression: $\frac{1}{√1+√2}$+$\frac{1} {√2 + √3}$+ $\frac{1}{√3 + √4}$+$\frac{1}{√2023+√2024}$

asked by guest
on Nov 16, 2024 at 12:20 am



You asked:

Simplify the expression: \(\frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{2023} + \sqrt{2024}}\)

MathBot Answer:

Simplified:

\[ \frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{2023} + \sqrt{2024}} = \frac{1}{17 \sqrt{7} + 2 \sqrt{506}} + \frac{1}{\sqrt{3} + 2} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{1 + \sqrt{2}} \]

Expanded:

\[ \frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{2023} + \sqrt{2024}} = \frac{1}{17 \sqrt{7} + 2 \sqrt{506}} + \frac{1}{\sqrt{3} + 2} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{1 + \sqrt{2}} \]

Factored:

\[ \frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \frac{1}{\sqrt{2023} + \sqrt{2024}} = \frac{4 \sqrt{3} + 7 + 5 \sqrt{2} + 3 \sqrt{6} + 68 \sqrt{21} + 8 \sqrt{1518} + 119 \sqrt{7} + 14 \sqrt{506} + 85 \sqrt{14} + 20 \sqrt{253} + 51 \sqrt{42} + 12 \sqrt{759}}{\left(1 + \sqrt{2}\right) \left(\sqrt{2} + \sqrt{3}\right) \left(\sqrt{3} + 2\right) \left(17 \sqrt{7} + 2 \sqrt{506}\right)} \]


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