Answer :
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.)
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.)
