Rationalize the denominator: \(\frac{9}{5-\sqrt{7}} \cdot \frac{5+\sqrt{7}}{5+\sqrt{7}}=\frac{45+9\sqrt{7}}{{5}^{2}-{\sqrt{7}}^{2}}\).
\[\frac{45+9\sqrt{7}}{{5}^{2}-{\sqrt{7}}^{2}}-\frac{22}{7+\sqrt{5}}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Factor out the common term \(9\).
\[\frac{9(5+\sqrt{7})}{{5}^{2}-{\sqrt{7}}^{2}}-\frac{22}{7+\sqrt{5}}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Simplify \({5}^{2}\) to \(25\).
\[\frac{9(5+\sqrt{7})}{25-{\sqrt{7}}^{2}}-\frac{22}{7+\sqrt{5}}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{9(5+\sqrt{7})}{25-7}-\frac{22}{7+\sqrt{5}}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Simplify \(25-7\) to \(18\).
\[\frac{9(5+\sqrt{7})}{18}-\frac{22}{7+\sqrt{5}}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Simplify \(\frac{9(5+\sqrt{7})}{18}\) to \(\frac{5+\sqrt{7}}{2}\).
\[\frac{5+\sqrt{7}}{2}-\frac{22}{7+\sqrt{5}}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Rationalize the denominator: \(\frac{22}{7+\sqrt{5}} \cdot \frac{7-\sqrt{5}}{7-\sqrt{5}}=\frac{154-22\sqrt{5}}{{7}^{2}-{\sqrt{5}}^{2}}\).
\[\frac{5+\sqrt{7}}{2}-\frac{154-22\sqrt{5}}{{7}^{2}-{\sqrt{5}}^{2}}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Factor out the common term \(22\).
\[\frac{5+\sqrt{7}}{2}-\frac{22(7-\sqrt{5})}{{7}^{2}-{\sqrt{5}}^{2}}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Simplify \({7}^{2}\) to \(49\).
\[\frac{5+\sqrt{7}}{2}-\frac{22(7-\sqrt{5})}{49-{\sqrt{5}}^{2}}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{5+\sqrt{7}}{2}-\frac{22(7-\sqrt{5})}{49-5}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Simplify \(49-5\) to \(44\).
\[\frac{5+\sqrt{7}}{2}-\frac{22(7-\sqrt{5})}{44}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Simplify \(\frac{22(7-\sqrt{5})}{44}\) to \(\frac{7-\sqrt{5}}{2}\).
\[\frac{5+\sqrt{7}}{2}-\frac{7-\sqrt{5}}{2}+\frac{1}{\sqrt{7}+\sqrt{5}}\]
Rationalize the denominator: \(\frac{1}{\sqrt{7}+\sqrt{5}} \cdot \frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}-\sqrt{5}}=\frac{\sqrt{7}-\sqrt{5}}{{\sqrt{7}}^{2}-{\sqrt{5}}^{2}}\).
\[\frac{5+\sqrt{7}}{2}-\frac{7-\sqrt{5}}{2}+\frac{\sqrt{7}-\sqrt{5}}{{\sqrt{7}}^{2}-{\sqrt{5}}^{2}}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{5+\sqrt{7}}{2}-\frac{7-\sqrt{5}}{2}+\frac{\sqrt{7}-\sqrt{5}}{7-{\sqrt{5}}^{2}}\]
Use this rule: \({\sqrt{x}}^{2}=x\).
\[\frac{5+\sqrt{7}}{2}-\frac{7-\sqrt{5}}{2}+\frac{\sqrt{7}-\sqrt{5}}{7-5}\]
Simplify \(7-5\) to \(2\).
\[\frac{5+\sqrt{7}}{2}-\frac{7-\sqrt{5}}{2}+\frac{\sqrt{7}-\sqrt{5}}{2}\]
Join the denominators.
\[\frac{5+\sqrt{7}-(7-\sqrt{5})+\sqrt{7}-\sqrt{5}}{2}\]
Remove parentheses.
\[\frac{5+\sqrt{7}-7+\sqrt{5}+\sqrt{7}-\sqrt{5}}{2}\]
Collect like terms.
\[\frac{(5-7)+(\sqrt{7}+\sqrt{7})+(\sqrt{5}-\sqrt{5})}{2}\]
Simplify \((5-7)+(\sqrt{7}+\sqrt{7})+(\sqrt{5}-\sqrt{5})\) to \(-2+2\sqrt{7}\).
\[\frac{-2+2\sqrt{7}}{2}\]
Factor out the common term \(2\).
\[\frac{-2(1-\sqrt{7})}{2}\]
Move the negative sign to the left.
\[-\frac{2(1-\sqrt{7})}{2}\]
Cancel \(2\).
\[-(1-\sqrt{7})\]
Remove parentheses.
\[-1+\sqrt{7}\]
Decimal Form: 1.645751
-1+sqrt(7)
