Solve frac{3sqrt{3}}{3sqrt{2}-2sqrt{3}}

$$\frac { 3 \sqrt { 3 } } { 3 \sqrt { 2 } - 2 \sqrt { 3 } }$$

Evaluate

$$\frac{3\left(\sqrt{6}+2\right)}{2}\approx 6.674234614$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{3}}{3\sqrt{2}-2\sqrt{3}} by multiplying numerator and denominator by 3\sqrt{2}+2\sqrt{3}.

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

Consider \left(3\sqrt{2}-2\sqrt{3}\right)\left(3\sqrt{2}+2\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}.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

The square of \sqrt{2} is 2.

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

Multiply 9 and 2 to get 18.

$$\frac{3\sqrt{3}\left(3\sqrt{2}+2\sqrt{3}\right)}{18-\left(-2\sqrt{3}\right)^{2}}$$

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

$$\frac{3\sqrt{3}\left(3\sqrt{2}+2\sqrt{3}\right)}{18-\left(-2\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -2 to the power of 2 and get 4.

$$\frac{3\sqrt{3}\left(3\sqrt{2}+2\sqrt{3}\right)}{18-4\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{3\sqrt{3}\left(3\sqrt{2}+2\sqrt{3}\right)}{18-4\times 3}$$

Multiply 4 and 3 to get 12.

$$\frac{3\sqrt{3}\left(3\sqrt{2}+2\sqrt{3}\right)}{18-12}$$

Subtract 12 from 18 to get 6.

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

Divide 3\sqrt{3}\left(3\sqrt{2}+2\sqrt{3}\right) by 6 to get \frac{1}{2}\sqrt{3}\left(3\sqrt{2}+2\sqrt{3}\right).

$$\frac{1}{2}\sqrt{3}\left(3\sqrt{2}+2\sqrt{3}\right)$$

Use the distributive property to multiply \frac{1}{2}\sqrt{3} by 3\sqrt{2}+2\sqrt{3}.

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

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

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

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

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

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

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

Multiply \frac{1}{2} and 3 to get \frac{3}{2}.

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

Cancel out 2 and 2.

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