Solve 5sqrt{45}*3/2sqrt{2/3}*sqrt{1/5}

$$5 \sqrt { 45 } \times \frac { 3 } { 2 } \sqrt { \frac { 2 } { 3 } } \times \sqrt { \frac { 1 } { 5 } }$$

Evaluate

$$\frac{15\sqrt{6}}{2}\approx 18.371173071$$

Solution Steps

To multiply \sqrt{\frac{2}{3}} and \sqrt{\frac{1}{5}}, multiply the numbers under the square root.

$$5\sqrt{45}\times \frac{3}{2}\sqrt{\frac{2}{15}}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

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

Multiply 5 and 3 to get 15.

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

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

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

Multiply 15 and 3 to get 45.

$$\frac{45}{2}\sqrt{5}\sqrt{\frac{2}{15}}$$

Rewrite the square root of the division \sqrt{\frac{2}{15}} as the division of square roots \frac{\sqrt{2}}{\sqrt{15}}.

$$\frac{45}{2}\sqrt{5}\times \frac{\sqrt{2}}{\sqrt{15}}$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$\frac{45}{2}\sqrt{5}\times \frac{\sqrt{2}\sqrt{15}}{\left(\sqrt{15}\right)^{2}}$$

The square of \sqrt{15} is 15.

$$\frac{45}{2}\sqrt{5}\times \frac{\sqrt{2}\sqrt{15}}{15}$$

To multiply \sqrt{2} and \sqrt{15}, multiply the numbers under the square root.

$$\frac{45}{2}\sqrt{5}\times \frac{\sqrt{30}}{15}$$

Multiply \frac{45}{2} times \frac{\sqrt{30}}{15} by multiplying numerator times numerator and denominator times denominator.

$$\frac{45\sqrt{30}}{2\times 15}\sqrt{5}$$

Cancel out 15 in both numerator and denominator.

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

Express \frac{3\sqrt{30}}{2}\sqrt{5} as a single fraction.

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

Factor 30=5\times 6. Rewrite the square root of the product \sqrt{5\times 6} as the product of square roots \sqrt{5}\sqrt{6}.

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

Multiply \sqrt{5} and \sqrt{5} to get 5.

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

Multiply 3 and 5 to get 15.

$$\frac{15\sqrt{6}}{2}$$