Solve sqrt{2/3}*sqrt{33/8}*sqrt{4/25}

$$\sqrt { \frac { 2 } { 3 } } \times \sqrt { 3 \frac { 3 } { 8 } } \cdot \sqrt { \frac { 4 } { 25 } }$$

Evaluate

$$\frac{3}{5}=0.6$$

Solution Steps

To multiply \sqrt{\frac{2}{3}} and \sqrt{\frac{4}{25}}, multiply the numbers under the square root.

$$\sqrt{\frac{8}{75}}\sqrt{\frac{3\times 8+3}{8}}$$

Rewrite the square root of the division \sqrt{\frac{8}{75}} as the division of square roots \frac{\sqrt{8}}{\sqrt{75}}.

$$\frac{\sqrt{8}}{\sqrt{75}}\sqrt{\frac{3\times 8+3}{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{2\sqrt{2}}{\sqrt{75}}\sqrt{\frac{3\times 8+3}{8}}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$\frac{2\sqrt{2}}{5\sqrt{3}}\sqrt{\frac{3\times 8+3}{8}}$$

Rationalize the denominator of \frac{2\sqrt{2}}{5\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{2\sqrt{2}\sqrt{3}}{5\left(\sqrt{3}\right)^{2}}\sqrt{\frac{3\times 8+3}{8}}$$

The square of \sqrt{3} is 3.

$$\frac{2\sqrt{2}\sqrt{3}}{5\times 3}\sqrt{\frac{3\times 8+3}{8}}$$

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

$$\frac{2\sqrt{6}}{5\times 3}\sqrt{\frac{3\times 8+3}{8}}$$

Multiply 5 and 3 to get 15.

$$\frac{2\sqrt{6}}{15}\sqrt{\frac{3\times 8+3}{8}}$$

Multiply 3 and 8 to get 24.

$$\frac{2\sqrt{6}}{15}\sqrt{\frac{24+3}{8}}$$

Add 24 and 3 to get 27.

$$\frac{2\sqrt{6}}{15}\sqrt{\frac{27}{8}}$$

Rewrite the square root of the division \sqrt{\frac{27}{8}} as the division of square roots \frac{\sqrt{27}}{\sqrt{8}}.

$$\frac{2\sqrt{6}}{15}\times \frac{\sqrt{27}}{\sqrt{8}}$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$\frac{2\sqrt{6}}{15}\times \frac{3\sqrt{3}}{\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

$$\frac{2\sqrt{6}}{15}\times \frac{3\sqrt{3}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

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

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

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

Multiply 2 and 2 to get 4.

$$\frac{2\sqrt{6}}{15}\times \frac{3\sqrt{6}}{4}$$

Multiply \frac{2\sqrt{6}}{15} times \frac{3\sqrt{6}}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2\sqrt{6}\times 3\sqrt{6}}{15\times 4}$$

Cancel out 2\times 3 in both numerator and denominator.

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

Multiply \sqrt{6} and \sqrt{6} to get 6.

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

Multiply 2 and 5 to get 10.

$$\frac{6}{10}$$

Reduce the fraction \frac{6}{10} to lowest terms by extracting and canceling out 2.

$$\frac{3}{5}$$

Factor

$$\frac{3}{5} = 0.6$$