Solve sqrt{1frac{2/3}}{sqrt{5/6}}

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

Evaluate

$$\sqrt{2}\approx 1.414213562$$

Solution Steps

Multiply 1 and 3 to get 3.

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

Add 3 and 2 to get 5.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

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

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

Rewrite the square root of the division \sqrt{\frac{5}{6}} as the division of square roots \frac{\sqrt{5}}{\sqrt{6}}.

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

Rationalize the denominator of \frac{\sqrt{5}}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{5}\sqrt{6}}{\left(\sqrt{6}\right)^{2}}}$$

The square of \sqrt{6} is 6.

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

To multiply \sqrt{5} and \sqrt{6}, multiply the numbers under the square root.

$$\frac{\frac{\sqrt{15}}{3}}{\frac{\sqrt{30}}{6}}$$

Divide \frac{\sqrt{15}}{3} by \frac{\sqrt{30}}{6} by multiplying \frac{\sqrt{15}}{3} by the reciprocal of \frac{\sqrt{30}}{6}.

$$\frac{\sqrt{15}\times 6}{3\sqrt{30}}$$

Cancel out 3 in both numerator and denominator.

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

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

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

The square of \sqrt{30} is 30.

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

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}.

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

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

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

Multiply 2 and 15 to get 30.

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

Cancel out 30 and 30.

$$\sqrt{2}$$