Solve 1/2sqrt{5/3}divsqrt{10/27}

$$\frac { 1 } { 2 } \sqrt { \frac { 5 } { 3 } } \div \sqrt { \frac { 10 } { 27 } }$$

Evaluate

$$\frac{3\sqrt{2}}{4}\approx 1.060660172$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{5}{3}} as the division of square roots \frac{\sqrt{5}}{\sqrt{3}}.

$$\frac{\frac{1}{2}\times \frac{\sqrt{5}}{\sqrt{3}}}{\sqrt{\frac{10}{27}}}$$

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

$$\frac{\frac{1}{2}\times \frac{\sqrt{5}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}}{\sqrt{\frac{10}{27}}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{1}{2}\times \frac{\sqrt{5}\sqrt{3}}{3}}{\sqrt{\frac{10}{27}}}$$

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

$$\frac{\frac{1}{2}\times \frac{\sqrt{15}}{3}}{\sqrt{\frac{10}{27}}}$$

Multiply \frac{1}{2} times \frac{\sqrt{15}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\frac{\sqrt{15}}{2\times 3}}{\sqrt{\frac{10}{27}}}$$

Rewrite the square root of the division \sqrt{\frac{10}{27}} as the division of square roots \frac{\sqrt{10}}{\sqrt{27}}.

$$\frac{\frac{\sqrt{15}}{2\times 3}}{\frac{\sqrt{10}}{\sqrt{27}}}$$

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}.

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

Rationalize the denominator of \frac{\sqrt{10}}{3\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{\frac{\sqrt{15}}{2\times 3}}{\frac{\sqrt{10}\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}}$$

The square of \sqrt{3} is 3.

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

To multiply \sqrt{10} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{15}}{2\times 3}}{\frac{\sqrt{30}}{3\times 3}}$$

Multiply 3 and 3 to get 9.

$$\frac{\frac{\sqrt{15}}{2\times 3}}{\frac{\sqrt{30}}{9}}$$

Divide \frac{\sqrt{15}}{2\times 3} by \frac{\sqrt{30}}{9} by multiplying \frac{\sqrt{15}}{2\times 3} by the reciprocal of \frac{\sqrt{30}}{9}.

$$\frac{\sqrt{15}\times 9}{2\times 3\sqrt{30}}$$

Cancel out 3 in both numerator and denominator.

$$\frac{3\sqrt{15}}{2\sqrt{30}}$$

Rationalize the denominator of \frac{3\sqrt{15}}{2\sqrt{30}} by multiplying numerator and denominator by \sqrt{30}.

$$\frac{3\sqrt{15}\sqrt{30}}{2\left(\sqrt{30}\right)^{2}}$$

The square of \sqrt{30} is 30.

$$\frac{3\sqrt{15}\sqrt{30}}{2\times 30}$$

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

$$\frac{3\sqrt{15}\sqrt{15}\sqrt{2}}{2\times 30}$$

Multiply \sqrt{15} and \sqrt{15} to get 15.

$$\frac{3\times 15\sqrt{2}}{2\times 30}$$

Multiply 2 and 30 to get 60.

$$\frac{3\times 15\sqrt{2}}{60}$$

Multiply 3 and 15 to get 45.

$$\frac{45\sqrt{2}}{60}$$

Divide 45\sqrt{2} by 60 to get \frac{3}{4}\sqrt{2}.

$$\frac{3}{4}\sqrt{2}$$