Solve frac{27(sqrt{36}-sqrt{30})}{3sqrt{2}}

$$\frac { 27 ( \sqrt { 36 } - \sqrt { 30 } ) } { 3 \sqrt { 2 } }$$

Evaluate

$$27\sqrt{2}-9\sqrt{15}\approx 3.326916068$$

Solution Steps

Cancel out 3 in both numerator and denominator.

$$\frac{9\left(\sqrt{36}-\sqrt{30}\right)}{\sqrt{2}}$$

Calculate the square root of 36 and get 6.

$$\frac{9\left(6-\sqrt{30}\right)}{\sqrt{2}}$$

Rationalize the denominator of \frac{9\left(6-\sqrt{30}\right)}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{9\left(6-\sqrt{30}\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{9\left(6-\sqrt{30}\right)\sqrt{2}}{2}$$

Use the distributive property to multiply 9 by 6-\sqrt{30}.

$$\frac{\left(54-9\sqrt{30}\right)\sqrt{2}}{2}$$

Use the distributive property to multiply 54-9\sqrt{30} by \sqrt{2}.

$$\frac{54\sqrt{2}-9\sqrt{30}\sqrt{2}}{2}$$

Factor 30=2\times 15. Rewrite the square root of the product \sqrt{2\times 15} as the product of square roots \sqrt{2}\sqrt{15}.

$$\frac{54\sqrt{2}-9\sqrt{2}\sqrt{15}\sqrt{2}}{2}$$

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

$$\frac{54\sqrt{2}-9\times 2\sqrt{15}}{2}$$

Multiply -9 and 2 to get -18.

$$\frac{54\sqrt{2}-18\sqrt{15}}{2}$$