Solve frac{6sqrt{2}}{sqrt{7}+2}

$$\frac { 6 \sqrt { 2 } } { \sqrt { 7 } + 2 }$$

Evaluate

$$2\sqrt{2}\left(\sqrt{7}-2\right)\approx 1.826460524$$

Solution Steps

Rationalize the denominator of \frac{6\sqrt{2}}{\sqrt{7}+2} by multiplying numerator and denominator by \sqrt{7}-2.

$$\frac{6\sqrt{2}\left(\sqrt{7}-2\right)}{\left(\sqrt{7}+2\right)\left(\sqrt{7}-2\right)}$$

Consider \left(\sqrt{7}+2\right)\left(\sqrt{7}-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}.

$$\frac{6\sqrt{2}\left(\sqrt{7}-2\right)}{\left(\sqrt{7}\right)^{2}-2^{2}}$$

Square \sqrt{7}. Square 2.

$$\frac{6\sqrt{2}\left(\sqrt{7}-2\right)}{7-4}$$

Subtract 4 from 7 to get 3.

$$\frac{6\sqrt{2}\left(\sqrt{7}-2\right)}{3}$$

Use the distributive property to multiply 6\sqrt{2} by \sqrt{7}-2.

$$\frac{6\sqrt{2}\sqrt{7}-12\sqrt{2}}{3}$$

To multiply \sqrt{2} and \sqrt{7}, multiply the numbers under the square root.

$$\frac{6\sqrt{14}-12\sqrt{2}}{3}$$