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

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

Evaluate

$$3\sqrt{3}\left(\sqrt{6}-2\right)\approx 2.335617216$$

Solution Steps

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

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

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

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

Square \sqrt{3}. Square \sqrt{2}.

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

Subtract 2 from 3 to get 1.

$$\frac{3\sqrt{6}\left(\sqrt{3}-\sqrt{2}\right)}{1}$$

Anything divided by one gives itself.

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

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

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

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}.

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

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

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

Multiply 3 and 3 to get 9.

$$9\sqrt{2}-3\sqrt{6}\sqrt{2}$$

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}.

$$9\sqrt{2}-3\sqrt{2}\sqrt{3}\sqrt{2}$$

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

$$9\sqrt{2}-3\times 2\sqrt{3}$$

Multiply -3 and 2 to get -6.

$$9\sqrt{2}-6\sqrt{3}$$