Solve 2/9-sqrt{15}

$$\frac { 2 } { 9 - \sqrt { 15 } }$$

Evaluate

$$\frac{\sqrt{15}+9}{33}\approx 0.390090404$$

Solution Steps

Rationalize the denominator of \frac{2}{9-\sqrt{15}} by multiplying numerator and denominator by 9+\sqrt{15}.

$$\frac{2\left(9+\sqrt{15}\right)}{\left(9-\sqrt{15}\right)\left(9+\sqrt{15}\right)}$$

Consider \left(9-\sqrt{15}\right)\left(9+\sqrt{15}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\left(9+\sqrt{15}\right)}{9^{2}-\left(\sqrt{15}\right)^{2}}$$

Square 9. Square \sqrt{15}.

$$\frac{2\left(9+\sqrt{15}\right)}{81-15}$$

Subtract 15 from 81 to get 66.

$$\frac{2\left(9+\sqrt{15}\right)}{66}$$

Divide 2\left(9+\sqrt{15}\right) by 66 to get \frac{1}{33}\left(9+\sqrt{15}\right).

$$\frac{1}{33}\left(9+\sqrt{15}\right)$$

Use the distributive property to multiply \frac{1}{33} by 9+\sqrt{15}.

$$\frac{1}{33}\times 9+\frac{1}{33}\sqrt{15}$$

Multiply \frac{1}{33} and 9 to get \frac{9}{33}.

$$\frac{9}{33}+\frac{1}{33}\sqrt{15}$$

Reduce the fraction \frac{9}{33} to lowest terms by extracting and canceling out 3.

$$\frac{3}{11}+\frac{1}{33}\sqrt{15}$$