Solve 2*frac{sqrt{12}}{5}*frac{sqrt{30}}{6sqrt{2}}

$$2 \cdot \frac { \sqrt { 12 } } { 5 } \cdot \frac { \sqrt { 30 } } { 6 \sqrt { 2 } }$$

Evaluate

$$\frac{2\sqrt{5}}{5}\approx 0.894427191$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$2\times \frac{2\sqrt{3}}{5}\times \frac{\sqrt{30}}{6\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{30}}{6\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$2\times \frac{2\sqrt{3}}{5}\times \frac{\sqrt{30}\sqrt{2}}{6\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$2\times \frac{2\sqrt{3}}{5}\times \frac{\sqrt{30}\sqrt{2}}{6\times 2}$$

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}.

$$2\times \frac{2\sqrt{3}}{5}\times \frac{\sqrt{2}\sqrt{15}\sqrt{2}}{6\times 2}$$

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

$$2\times \frac{2\sqrt{3}}{5}\times \frac{2\sqrt{15}}{6\times 2}$$

Multiply 6 and 2 to get 12.

$$2\times \frac{2\sqrt{3}}{5}\times \frac{2\sqrt{15}}{12}$$

Divide 2\sqrt{15} by 12 to get \frac{1}{6}\sqrt{15}.

$$2\times \frac{2\sqrt{3}}{5}\times \frac{1}{6}\sqrt{15}$$

Multiply 2 and \frac{1}{6} to get \frac{2}{6}.

$$\frac{2}{6}\times \frac{2\sqrt{3}}{5}\sqrt{15}$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$\frac{1}{3}\times \frac{2\sqrt{3}}{5}\sqrt{15}$$

Multiply \frac{1}{3} times \frac{2\sqrt{3}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\sqrt{3}}{3\times 5}\sqrt{15}$$

Express \frac{2\sqrt{3}}{3\times 5}\sqrt{15} as a single fraction.

$$\frac{2\sqrt{3}\sqrt{15}}{3\times 5}$$

Factor 15=3\times 5. Rewrite the square root of the product \sqrt{3\times 5} as the product of square roots \sqrt{3}\sqrt{5}.

$$\frac{2\sqrt{3}\sqrt{3}\sqrt{5}}{3\times 5}$$

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

$$\frac{2\times 3\sqrt{5}}{3\times 5}$$

Multiply 2 and 3 to get 6.

$$\frac{6\sqrt{5}}{3\times 5}$$

Multiply 3 and 5 to get 15.

$$\frac{6\sqrt{5}}{15}$$

Divide 6\sqrt{5} by 15 to get \frac{2}{5}\sqrt{5}.

$$\frac{2}{5}\sqrt{5}$$