Rationalize the denominator: \(\frac{3}{7+3\sqrt{2}} \cdot \frac{7-3\sqrt{2}}{7-3\sqrt{2}}=\frac{21-9\sqrt{2}}{{7}^{2}-{(3\sqrt{2})}^{2}}\).
\[\frac{21-9\sqrt{2}}{{7}^{2}-{(3\sqrt{2})}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Factor out the common term \(3\).
\[\frac{3(7-3\sqrt{2})}{{7}^{2}-{(3\sqrt{2})}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Simplify \({7}^{2}\) to \(49\).
\[\frac{3(7-3\sqrt{2})}{49-{(3\sqrt{2})}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Rationalize the denominator: \(\frac{3(7-3\sqrt{2})}{49-{(3\sqrt{2})}^{2}} \cdot \frac{49+{(3\sqrt{2})}^{2}}{49+{(3\sqrt{2})}^{2}}=\frac{1029+378-441\sqrt{2}-162\sqrt{2}}{{49}^{2}-{({(3\sqrt{2})}^{2})}^{2}}\).
\[\frac{1029+378-441\sqrt{2}-162\sqrt{2}}{{49}^{2}-{({(3\sqrt{2})}^{2})}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Factor out the common term \(3\).
\[\frac{3(343+126-147\sqrt{2}-54\sqrt{2})}{{49}^{2}-{({(3\sqrt{2})}^{2})}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Collect like terms.
\[\frac{3((343+126)+(-147\sqrt{2}-54\sqrt{2}))}{{49}^{2}-{({(3\sqrt{2})}^{2})}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Simplify \((343+126)+(-147\sqrt{2}-54\sqrt{2})\) to \(469-201\sqrt{2}\).
\[\frac{3(469-201\sqrt{2})}{{49}^{2}-{({(3\sqrt{2})}^{2})}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Factor out the common term \(67\).
\[\frac{3\times 67(7-3\sqrt{2})}{{49}^{2}-{({(3\sqrt{2})}^{2})}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Simplify \(3\times 67(7-3\sqrt{2})\) to \(201(7-3\sqrt{2})\).
\[\frac{201(7-3\sqrt{2})}{{49}^{2}-{({(3\sqrt{2})}^{2})}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Simplify \({49}^{2}\) to \(2401\).
\[\frac{201(7-3\sqrt{2})}{2401-{({(3\sqrt{2})}^{2})}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Use Power Rule: \({({x}^{a})}^{b}={x}^{ab}\).
\[\frac{201(7-3\sqrt{2})}{2401-{(3\sqrt{2})}^{4}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Rewrite \(2401-{(3\sqrt{2})}^{4}\) in the form \({a}^{2}-{b}^{2}\), where \(a=49\) and \(b=18\).
\[\frac{201(7-3\sqrt{2})}{{49}^{2}-{18}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Simplify \({49}^{2}\) to \(2401\).
\[\frac{201(7-3\sqrt{2})}{2401-{18}^{2}}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Simplify \({18}^{2}\) to \(324\).
\[\frac{201(7-3\sqrt{2})}{2401-324}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Simplify \(2401-324\) to \(2077\).
\[\frac{201(7-3\sqrt{2})}{2077}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Simplify \(\frac{201(7-3\sqrt{2})}{2077}\) to \(\frac{3(7-3\sqrt{2})}{31}\).
\[\frac{3(7-3\sqrt{2})}{31}\times \frac{7-3\sqrt{2}}{7-3\sqrt{2}}\]
Cancel \(7-3\sqrt{2}\).
\[\frac{3}{31}(7-3\sqrt{2})\]
Simplify.
\[\frac{3(7-3\sqrt{2})}{31}\]
Decimal Form: 0.266841
(3*(7-3*sqrt(2)))/31
