Solve sqrt{2/45}div3/2sqrt{8/5}

$$\sqrt { \frac { 2 } { 45 } } \div \frac { 3 } { 2 } \sqrt { \frac { 8 } { 5 } }$$

Evaluate

$$\frac{8}{45}\approx 0.177777778$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{2}{45}} as the division of square roots \frac{\sqrt{2}}{\sqrt{45}}.

$$\frac{\frac{\sqrt{2}}{\sqrt{45}}}{\frac{3}{2}}\sqrt{\frac{8}{5}}$$

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}.

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

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

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

The square of \sqrt{5} is 5.

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

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

$$\frac{\frac{\sqrt{10}}{3\times 5}}{\frac{3}{2}}\sqrt{\frac{8}{5}}$$

Multiply 3 and 5 to get 15.

$$\frac{\frac{\sqrt{10}}{15}}{\frac{3}{2}}\sqrt{\frac{8}{5}}$$

Divide \frac{\sqrt{10}}{15} by \frac{3}{2} by multiplying \frac{\sqrt{10}}{15} by the reciprocal of \frac{3}{2}.

$$\frac{\sqrt{10}\times 2}{15\times 3}\sqrt{\frac{8}{5}}$$

Multiply 15 and 3 to get 45.

$$\frac{\sqrt{10}\times 2}{45}\sqrt{\frac{8}{5}}$$

Rewrite the square root of the division \sqrt{\frac{8}{5}} as the division of square roots \frac{\sqrt{8}}{\sqrt{5}}.

$$\frac{\sqrt{10}\times 2}{45}\times \frac{\sqrt{8}}{\sqrt{5}}$$

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}.

$$\frac{\sqrt{10}\times 2}{45}\times \frac{2\sqrt{2}}{\sqrt{5}}$$

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

$$\frac{\sqrt{10}\times 2}{45}\times \frac{2\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\sqrt{10}\times 2}{45}\times \frac{2\sqrt{2}\sqrt{5}}{5}$$

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

$$\frac{\sqrt{10}\times 2}{45}\times \frac{2\sqrt{10}}{5}$$

Multiply \frac{\sqrt{10}\times 2}{45} times \frac{2\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\sqrt{10}\times 2\times 2\sqrt{10}}{45\times 5}$$

Multiply \sqrt{10} and \sqrt{10} to get 10.

$$\frac{10\times 2\times 2}{45\times 5}$$

Cancel out 5 in both numerator and denominator.

$$\frac{2\times 2\times 2}{5\times 9}$$

Multiply 2 and 2 to get 4.

$$\frac{4\times 2}{5\times 9}$$

Multiply 4 and 2 to get 8.

$$\frac{8}{5\times 9}$$

Multiply 5 and 9 to get 45.

$$\frac{8}{45}$$

Factor

$$\frac{2 ^ {3}}{3 ^ {2} \cdot 5} = 0.17777777777777778$$