To simplify the expression [tex]\(-\frac{1}{2 + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}}\)[/tex], let's break it down and simplify each part step by step.
First, we have the expression:
[tex]\[
-\frac{1}{2 + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}}
\][/tex]
### Simplifying Each Term:
1. Rationalizing [tex]\(\frac{-1}{2+\sqrt{5}}\)[/tex]:
Multiply numerator and denominator by the conjugate of the denominator (i.e., [tex]\(2-\sqrt{5}\)[/tex]):
[tex]\[
-\frac{1}{2 + \sqrt{5}} \cdot \frac{2 - \sqrt{5}}{2 - \sqrt{5}} = -\frac{2 - \sqrt{5}}{(2 + \sqrt{5})(2 - \sqrt{5})} = -\frac{2 - \sqrt{5}}{4 - 5} = -\frac{2 - \sqrt{5}}{-1} = 2 - \sqrt{5}
\][/tex]
2. Rationalizing [tex]\(\frac{1}{\sqrt{5}+\sqrt{6}}\)[/tex]:
Multiply numerator and denominator by the conjugate of the denominator (i.e., [tex]\(\sqrt{5}-\sqrt{6}\)[/tex]):
[tex]\[
\frac{1}{\sqrt{5} + \sqrt{6}} \cdot \frac{\sqrt{5} - \sqrt{6}}{\sqrt{5} - \sqrt{6}} = \frac{\sqrt{5} - \sqrt{6}}{(\sqrt{5})^2 - (\sqrt{6})^2} = \frac{\sqrt{5} - \sqrt{6}}{5 - 6} = -(\sqrt{5} - \sqrt{6})
\][/tex]
3. Rationalizing [tex]\(\frac{1}{\sqrt{6}+\sqrt{7}}\)[/tex]:
Multiply numerator and denominator by the conjugate of the denominator (i.e., [tex]\(\sqrt{6}-\sqrt{7}\)[/tex]):
[tex]\[
\frac{1}{\sqrt{6} + \sqrt{7}} \cdot \frac{\sqrt{6} - \sqrt{7}}{\sqrt{6} - \sqrt{7}} = \frac{\sqrt{6} - \sqrt{7}}{(\sqrt{6})^2 - (\sqrt{7})^2} = \frac{\sqrt{6} - \sqrt{7}}{6 - 7} = -(\sqrt{6} - \sqrt{7})
\][/tex]
4. Rationalizing [tex]\(\frac{1}{\sqrt{7}+\sqrt{8}}\)[/tex]:
Multiply numerator and denominator by the conjugate of the denominator (i.e., [tex]\(\sqrt{7}-\sqrt{8}\)[/tex]):
[tex]\[
\frac{1}{\sqrt{7} + \sqrt{8}} \cdot \frac{\sqrt{7} - \sqrt{8}}{\sqrt{7} - \sqrt{8}} = \frac{\sqrt{7} - \sqrt{8}}{(\sqrt{7})^2 - (\sqrt{8})^2} = \frac{\sqrt{7} - \sqrt{8}}{7 - 8} = -(\sqrt{7} - \sqrt{8})
\][/tex]
### Simplifying the Expression Together:
Now, combining all the rationalized parts:
[tex]\[
2 - \sqrt{5} -(\sqrt{5} - \sqrt{6}) -(\sqrt{6} - \sqrt{7}) -(\sqrt{7} - \sqrt{8})
\][/tex]
Simplify inside the parentheses:
[tex]\(
-\sqrt{5} + \sqrt{6} - \sqrt{6} + \sqrt{7} - \sqrt{7} + \sqrt{8}
\)[/tex]
Thus:
[tex]\[
2 - \sqrt{5} - \sqrt{5} + \sqrt{6} - \sqrt{6} + \sqrt{7} - \sqrt{7} + \sqrt{8}
\][/tex]
Further simplification and combining like terms shows the original simplified expression:
[tex]\[
-\frac{1}{2+\sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}}
\][/tex]
is indeed:
[tex]\[
-\frac{1}{2 + \sqrt{5}} + \frac{1}{\sqrt{7} + 2 \sqrt{2}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{5} + \sqrt{6}}
\][/tex]
So the simplified form remains:
[tex]\[
-\frac{1}{2 + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{6}} + \frac{1}{\sqrt{6} + \sqrt{7}} + \frac{1}{\sqrt{7} + \sqrt{8}}
\][/tex]
(Alert: Actual simplification might have deeper steps and involve understanding the ultimate value which in some cases retain original structure post conjugation handling as symmetry in terms.)
