Solve frac{3sqrt{2}}{sqrt{6}-sqrt{3}}

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

Evaluate

$$\sqrt{6}+2\sqrt{3}\approx 5.913591358$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{2}}{\sqrt{6}-\sqrt{3}} by multiplying numerator and denominator by \sqrt{6}+\sqrt{3}.

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

Consider \left(\sqrt{6}-\sqrt{3}\right)\left(\sqrt{6}+\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}.

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

Square \sqrt{6}. Square \sqrt{3}.

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

Subtract 3 from 6 to get 3.

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

Use the distributive property to multiply 3\sqrt{2} by \sqrt{6}+\sqrt{3}.

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

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

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Multiply 3 and 2 to get 6.

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

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

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

Divide each term of 6\sqrt{3}+3\sqrt{6} by 3 to get 2\sqrt{3}+\sqrt{6}.

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