Solve sqrt{3/2}divsqrt{1/8}

$$\sqrt { \frac { 3 } { 2 } } \div \sqrt { \frac { 1 } { 8 } }$$

Evaluate

$$2\sqrt{3}\approx 3.464101615$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

$$\frac{\frac{\sqrt{3}}{\sqrt{2}}}{\sqrt{\frac{1}{8}}}$$

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

$$\frac{\frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}{\sqrt{\frac{1}{8}}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{\sqrt{3}\sqrt{2}}{2}}{\sqrt{\frac{1}{8}}}$$

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

$$\frac{\frac{\sqrt{6}}{2}}{\sqrt{\frac{1}{8}}}$$

Rewrite the square root of the division \sqrt{\frac{1}{8}} as the division of square roots \frac{\sqrt{1}}{\sqrt{8}}.

$$\frac{\frac{\sqrt{6}}{2}}{\frac{\sqrt{1}}{\sqrt{8}}}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{\sqrt{6}}{2}}{\frac{1}{\sqrt{8}}}$$

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}.

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

Rationalize the denominator of \frac{1}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

$$\frac{\frac{\sqrt{6}}{2}}{\frac{\sqrt{2}}{4}}$$

Divide \frac{\sqrt{6}}{2} by \frac{\sqrt{2}}{4} by multiplying \frac{\sqrt{6}}{2} by the reciprocal of \frac{\sqrt{2}}{4}.

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

Cancel out 2 in both numerator and denominator.

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

Rationalize the denominator of \frac{2\sqrt{6}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

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

Multiply \sqrt{2} and \sqrt{2} to get 2.

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

Cancel out 2 and 2.

$$2\sqrt{3}$$