Solve frac{3sqrt{6}}{6-3sqrt{3}}

$$\frac { 3 \sqrt { 6 } } { 6 - 3 \sqrt { 3 } }$$

Evaluate

$$\sqrt{2}\left(2\sqrt{3}+3\right)\approx 9.141620173$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{6}}{6-3\sqrt{3}} by multiplying numerator and denominator by 6+3\sqrt{3}.

$$\frac{3\sqrt{6}\left(6+3\sqrt{3}\right)}{\left(6-3\sqrt{3}\right)\left(6+3\sqrt{3}\right)}$$

Consider \left(6-3\sqrt{3}\right)\left(6+3\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{3\sqrt{6}\left(6+3\sqrt{3}\right)}{6^{2}-\left(-3\sqrt{3}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{3\sqrt{6}\left(6+3\sqrt{3}\right)}{36-\left(-3\sqrt{3}\right)^{2}}$$

Expand \left(-3\sqrt{3}\right)^{2}.

$$\frac{3\sqrt{6}\left(6+3\sqrt{3}\right)}{36-\left(-3\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{3\sqrt{6}\left(6+3\sqrt{3}\right)}{36-9\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3\sqrt{6}\left(6+3\sqrt{3}\right)}{36-9\times 3}$$

Multiply 9 and 3 to get 27.

$$\frac{3\sqrt{6}\left(6+3\sqrt{3}\right)}{36-27}$$

Subtract 27 from 36 to get 9.

$$\frac{3\sqrt{6}\left(6+3\sqrt{3}\right)}{9}$$

Divide 3\sqrt{6}\left(6+3\sqrt{3}\right) by 9 to get \frac{1}{3}\sqrt{6}\left(6+3\sqrt{3}\right).

$$\frac{1}{3}\sqrt{6}\left(6+3\sqrt{3}\right)$$

Use the distributive property to multiply \frac{1}{3}\sqrt{6} by 6+3\sqrt{3}.

$$\frac{1}{3}\sqrt{6}\times 6+\frac{1}{3}\sqrt{6}\times 3\sqrt{3}$$

Multiply \frac{1}{3} and 6 to get \frac{6}{3}.

$$\frac{6}{3}\sqrt{6}+\frac{1}{3}\sqrt{6}\times 3\sqrt{3}$$

Divide 6 by 3 to get 2.

$$2\sqrt{6}+\frac{1}{3}\sqrt{6}\times 3\sqrt{3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$2\sqrt{6}+\frac{1}{3}\sqrt{3}\sqrt{2}\times 3\sqrt{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$2\sqrt{6}+\frac{1}{3}\times 3\times 3\sqrt{2}$$

Cancel out 3 and 3.

$$2\sqrt{6}+3\sqrt{2}$$