Rationalize the denominator: \(\frac{1}{3\sqrt{5}+2\sqrt{2}} \cdot \frac{3\sqrt{5}-2\sqrt{2}}{3\sqrt{5}-2\sqrt{2}}=\frac{3\sqrt{5}-2\sqrt{2}}{{(3\sqrt{5})}^{2}-{(2\sqrt{2})}^{2}}\).
\[\frac{3\sqrt{5}-2\sqrt{2}}{{(3\sqrt{5})}^{2}-{(2\sqrt{2})}^{2}}\]
Rationalize the denominator: \(\frac{3\sqrt{5}-2\sqrt{2}}{{(3\sqrt{5})}^{2}-{(2\sqrt{2})}^{2}} \cdot \frac{{(3\sqrt{5})}^{2}+{(2\sqrt{2})}^{2}}{{(3\sqrt{5})}^{2}+{(2\sqrt{2})}^{2}}=\frac{135\sqrt{5}+24\sqrt{5}-90\sqrt{2}-16\sqrt{2}}{{({(3\sqrt{5})}^{2})}^{2}-{({(2\sqrt{2})}^{2})}^{2}}\).
\[\frac{135\sqrt{5}+24\sqrt{5}-90\sqrt{2}-16\sqrt{2}}{{({(3\sqrt{5})}^{2})}^{2}-{({(2\sqrt{2})}^{2})}^{2}}\]
Collect like terms.
\[\frac{(135\sqrt{5}+24\sqrt{5})+(-90\sqrt{2}-16\sqrt{2})}{{({(3\sqrt{5})}^{2})}^{2}-{({(2\sqrt{2})}^{2})}^{2}}\]
Simplify \((135\sqrt{5}+24\sqrt{5})+(-90\sqrt{2}-16\sqrt{2})\) to \(159\sqrt{5}-106\sqrt{2}\).
\[\frac{159\sqrt{5}-106\sqrt{2}}{{({(3\sqrt{5})}^{2})}^{2}-{({(2\sqrt{2})}^{2})}^{2}}\]
Factor out the common term \(53\).
\[\frac{53(3\sqrt{5}-2\sqrt{2})}{{({(3\sqrt{5})}^{2})}^{2}-{({(2\sqrt{2})}^{2})}^{2}}\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{53(3\sqrt{5}-2\sqrt{2})}{{(3\sqrt{5})}^{4}-{({(2\sqrt{2})}^{2})}^{2}}\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{53(3\sqrt{5}-2\sqrt{2})}{{(3\sqrt{5})}^{4}-{(2\sqrt{2})}^{4}}\]
Rewrite \({(3\sqrt{5})}^{4}-{(2\sqrt{2})}^{4}\) in the form \({a}^{2}-{b}^{2}\), where \(a=45\) and \(b=8\).
\[\frac{53(3\sqrt{5}-2\sqrt{2})}{{45}^{2}-{8}^{2}}\]
Simplify \({45}^{2}\) to \(2025\).
\[\frac{53(3\sqrt{5}-2\sqrt{2})}{2025-{8}^{2}}\]
Simplify \({8}^{2}\) to \(64\).
\[\frac{53(3\sqrt{5}-2\sqrt{2})}{2025-64}\]
Simplify \(2025-64\) to \(1961\).
\[\frac{53(3\sqrt{5}-2\sqrt{2})}{1961}\]
Simplify.
\[\frac{3\sqrt{5}-2\sqrt{2}}{37}\]
Decimal Form: 0.104859
(3*sqrt(5)-2*sqrt(2))/37
