Evaluate
\frac{63-7\sqrt{13}}{68}\approx 0.555310898
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\frac{7\left(9-\sqrt{13}\right)}{\left(9+\sqrt{13}\right)\left(9-\sqrt{13}\right)}
Rationalize the denominator of \frac{7}{9+\sqrt{13}} by multiplying numerator and denominator by 9-\sqrt{13}.
\frac{7\left(9-\sqrt{13}\right)}{9^{2}-\left(\sqrt{13}\right)^{2}}
Consider \left(9+\sqrt{13}\right)\left(9-\sqrt{13}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{7\left(9-\sqrt{13}\right)}{81-13}
Square 9. Square \sqrt{13}.
\frac{7\left(9-\sqrt{13}\right)}{68}
Subtract 13 from 81 to get 68.
\frac{63-7\sqrt{13}}{68}
Use the distributive property to multiply 7 by 9-\sqrt{13}.
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