Since \(2\times 2=4\), the square root of \(4\) is \(2\).
\[\frac{1}{2+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+\sqrt{8}}+\frac{1}{\sqrt{8}+\sqrt{9}}\]
Simplify \(\sqrt{8}\) to \(2\sqrt{2}\).
\[\frac{1}{2+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+\sqrt{9}}\]
Since \(3\times 3=9\), the square root of \(9\) is \(3\).
\[\frac{1}{2+\sqrt{5}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Rationalize the denominator: \(\frac{1}{2+\sqrt{5}} \cdot \frac{2-\sqrt{5}}{2-\sqrt{5}}=\frac{2-\sqrt{5}}{{2}^{2}-{\sqrt{5}}^{2}}\).
\[\frac{2-\sqrt{5}}{{2}^{2}-{\sqrt{5}}^{2}}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{2-\sqrt{5}}{{2}^{2}-5}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Convert to common denominators.
\[\frac{2-\sqrt{5}}{-1}+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Move the negative sign to the left.
\[-(2-\sqrt{5})+\frac{1}{\sqrt{5}+\sqrt{6}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Rationalize the denominator: \(\frac{1}{\sqrt{5}+\sqrt{6}} \cdot \frac{\sqrt{5}-\sqrt{6}}{\sqrt{5}-\sqrt{6}}=\frac{\sqrt{5}-\sqrt{6}}{{\sqrt{5}}^{2}-{\sqrt{6}}^{2}}\).
\[-(2-\sqrt{5})+\frac{\sqrt{5}-\sqrt{6}}{{\sqrt{5}}^{2}-{\sqrt{6}}^{2}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[-(2-\sqrt{5})+\frac{\sqrt{5}-\sqrt{6}}{5-{\sqrt{6}}^{2}}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[-(2-\sqrt{5})+\frac{\sqrt{5}-\sqrt{6}}{5-6}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Simplify \(5-6\) to \(-1\).
\[-(2-\sqrt{5})+\frac{\sqrt{5}-\sqrt{6}}{-1}+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Move the negative sign to the left.
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})+\frac{1}{\sqrt{6}+\sqrt{7}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Rationalize the denominator: \(\frac{1}{\sqrt{6}+\sqrt{7}} \cdot \frac{\sqrt{6}-\sqrt{7}}{\sqrt{6}-\sqrt{7}}=\frac{\sqrt{6}-\sqrt{7}}{{\sqrt{6}}^{2}-{\sqrt{7}}^{2}}\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})+\frac{\sqrt{6}-\sqrt{7}}{{\sqrt{6}}^{2}-{\sqrt{7}}^{2}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})+\frac{\sqrt{6}-\sqrt{7}}{6-{\sqrt{7}}^{2}}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})+\frac{\sqrt{6}-\sqrt{7}}{6-7}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Simplify \(6-7\) to \(-1\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})+\frac{\sqrt{6}-\sqrt{7}}{-1}+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Move the negative sign to the left.
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{1}{\sqrt{7}+2\sqrt{2}}+\frac{1}{2\sqrt{2}+3}\]
Rationalize the denominator: \(\frac{1}{\sqrt{7}+2\sqrt{2}} \cdot \frac{\sqrt{7}-2\sqrt{2}}{\sqrt{7}-2\sqrt{2}}=\frac{\sqrt{7}-2\sqrt{2}}{{\sqrt{7}}^{2}-{(2\sqrt{2})}^{2}}\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{\sqrt{7}-2\sqrt{2}}{{\sqrt{7}}^{2}-{(2\sqrt{2})}^{2}}+\frac{1}{2\sqrt{2}+3}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{\sqrt{7}-2\sqrt{2}}{7-{(2\sqrt{2})}^{2}}+\frac{1}{2\sqrt{2}+3}\]
Rationalize the denominator: \(\frac{\sqrt{7}-2\sqrt{2}}{7-{(2\sqrt{2})}^{2}} \cdot \frac{7+{(2\sqrt{2})}^{2}}{7+{(2\sqrt{2})}^{2}}=\frac{7\sqrt{7}+8\sqrt{7}-14\sqrt{2}-16\sqrt{2}}{{7}^{2}-{({(2\sqrt{2})}^{2})}^{2}}\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{7\sqrt{7}+8\sqrt{7}-14\sqrt{2}-16\sqrt{2}}{{7}^{2}-{({(2\sqrt{2})}^{2})}^{2}}+\frac{1}{2\sqrt{2}+3}\]
Collect like terms.
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{(7\sqrt{7}+8\sqrt{7})+(-14\sqrt{2}-16\sqrt{2})}{{7}^{2}-{({(2\sqrt{2})}^{2})}^{2}}+\frac{1}{2\sqrt{2}+3}\]
Simplify \((7\sqrt{7}+8\sqrt{7})+(-14\sqrt{2}-16\sqrt{2})\) to \(15\sqrt{7}-30\sqrt{2}\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{15\sqrt{7}-30\sqrt{2}}{{7}^{2}-{({(2\sqrt{2})}^{2})}^{2}}+\frac{1}{2\sqrt{2}+3}\]
Factor out the common term \(15\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{15(\sqrt{7}-2\sqrt{2})}{{7}^{2}-{({(2\sqrt{2})}^{2})}^{2}}+\frac{1}{2\sqrt{2}+3}\]
Simplify \({7}^{2}\) to \(49\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{15(\sqrt{7}-2\sqrt{2})}{49-{({(2\sqrt{2})}^{2})}^{2}}+\frac{1}{2\sqrt{2}+3}\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{15(\sqrt{7}-2\sqrt{2})}{49-{(2\sqrt{2})}^{4}}+\frac{1}{2\sqrt{2}+3}\]
Rewrite \(49-{(2\sqrt{2})}^{4}\) in the form \({a}^{2}-{b}^{2}\), where \(a=7\) and \(b=8\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{15(\sqrt{7}-2\sqrt{2})}{{7}^{2}-{8}^{2}}+\frac{1}{2\sqrt{2}+3}\]
Simplify \({7}^{2}\) to \(49\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{15(\sqrt{7}-2\sqrt{2})}{49-{8}^{2}}+\frac{1}{2\sqrt{2}+3}\]
Simplify \({8}^{2}\) to \(64\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{15(\sqrt{7}-2\sqrt{2})}{49-64}+\frac{1}{2\sqrt{2}+3}\]
Simplify \(49-64\) to \(-15\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})+\frac{15(\sqrt{7}-2\sqrt{2})}{-15}+\frac{1}{2\sqrt{2}+3}\]
Move the negative sign to the left.
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-\frac{15(\sqrt{7}-2\sqrt{2})}{15}+\frac{1}{2\sqrt{2}+3}\]
Cancel \(15\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{1}{2\sqrt{2}+3}\]
Rationalize the denominator: \(\frac{1}{2\sqrt{2}+3} \cdot \frac{2\sqrt{2}-3}{2\sqrt{2}-3}=\frac{2\sqrt{2}-3}{{(2\sqrt{2})}^{2}-{3}^{2}}\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{2\sqrt{2}-3}{{(2\sqrt{2})}^{2}-{3}^{2}}\]
Simplify \({3}^{2}\) to \(9\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{2\sqrt{2}-3}{{(2\sqrt{2})}^{2}-9}\]
Rationalize the denominator: \(\frac{2\sqrt{2}-3}{{(2\sqrt{2})}^{2}-9} \cdot \frac{{(2\sqrt{2})}^{2}+9}{{(2\sqrt{2})}^{2}+9}=\frac{16\sqrt{2}+18\sqrt{2}-24-27}{{({(2\sqrt{2})}^{2})}^{2}-{9}^{2}}\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{16\sqrt{2}+18\sqrt{2}-24-27}{{({(2\sqrt{2})}^{2})}^{2}-{9}^{2}}\]
Collect like terms.
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{(16\sqrt{2}+18\sqrt{2})+(-24-27)}{{({(2\sqrt{2})}^{2})}^{2}-{9}^{2}}\]
Simplify \((16\sqrt{2}+18\sqrt{2})+(-24-27)\) to \(34\sqrt{2}-51\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{34\sqrt{2}-51}{{({(2\sqrt{2})}^{2})}^{2}-{9}^{2}}\]
Factor out the common term \(17\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{17(2\sqrt{2}-3)}{{({(2\sqrt{2})}^{2})}^{2}-{9}^{2}}\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{17(2\sqrt{2}-3)}{{(2\sqrt{2})}^{4}-{9}^{2}}\]
Simplify \({9}^{2}\) to \(81\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{17(2\sqrt{2}-3)}{{(2\sqrt{2})}^{4}-81}\]
Rewrite \({(2\sqrt{2})}^{4}-81\) in the form \({a}^{2}-{b}^{2}\), where \(a=8\) and \(b=9\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{17(2\sqrt{2}-3)}{{8}^{2}-{9}^{2}}\]
Simplify \({8}^{2}\) to \(64\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{17(2\sqrt{2}-3)}{64-{9}^{2}}\]
Simplify \({9}^{2}\) to \(81\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{17(2\sqrt{2}-3)}{64-81}\]
Simplify \(64-81\) to \(-17\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})+\frac{17(2\sqrt{2}-3)}{-17}\]
Move the negative sign to the left.
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})-\frac{17(2\sqrt{2}-3)}{17}\]
Cancel \(17\).
\[-(2-\sqrt{5})-(\sqrt{5}-\sqrt{6})-(\sqrt{6}-\sqrt{7})-(\sqrt{7}-2\sqrt{2})-(2\sqrt{2}-3)\]
Remove parentheses.
\[-2+\sqrt{5}-\sqrt{5}+\sqrt{6}-\sqrt{6}+\sqrt{7}-\sqrt{7}+2\sqrt{2}-2\sqrt{2}+3\]
Collect like terms.
\[(-2+3)+(\sqrt{5}-\sqrt{5})+(\sqrt{6}-\sqrt{6})+(\sqrt{7}-\sqrt{7})+(2\sqrt{2}-2\sqrt{2})\]
Simplify.
\[1\]
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