Solve frac{12sqrt{2}}{sqrt{3}+sqrt{2}}

$$\frac { 12 \sqrt { 2 } } { \sqrt { 3 } + \sqrt { 2 } }$$

Evaluate

$$12\left(\sqrt{6}-2\right)\approx 5.393876913$$

Solution Steps

Rationalize the denominator of \frac{12\sqrt{2}}{\sqrt{3}+\sqrt{2}} by multiplying numerator and denominator by \sqrt{3}-\sqrt{2}.

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

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}.

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

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

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

Subtract 2 from 3 to get 1.

$$\frac{12\sqrt{2}\left(\sqrt{3}-\sqrt{2}\right)}{1}$$

Anything divided by one gives itself.

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

Use the distributive property to multiply 12\sqrt{2} by \sqrt{3}-\sqrt{2}.

$$12\sqrt{2}\sqrt{3}-12\left(\sqrt{2}\right)^{2}$$

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

$$12\sqrt{6}-12\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$12\sqrt{6}-12\times 2$$

Multiply -12 and 2 to get -24.

$$12\sqrt{6}-24$$