Solve 4sqrt{3}*frac{sqrt{3}}{4}*1/5sqrt{1/2}

$$4 \sqrt { 3 } \times \frac { \sqrt { 3 } } { 4 } \times \frac { 1 } { 5 } \sqrt { \frac { 1 } { 2 } }$$

Evaluate

$$\frac{3\sqrt{2}}{10}\approx 0.424264069$$

Solution Steps

Multiply 4 and \frac{1}{5} to get \frac{4}{5}.

$$\frac{4}{5}\sqrt{3}\times \frac{\sqrt{3}}{4}\sqrt{\frac{1}{2}}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$\frac{4}{5}\sqrt{3}\times \frac{\sqrt{3}}{4}\times \frac{\sqrt{1}}{\sqrt{2}}$$

Calculate the square root of 1 and get 1.

$$\frac{4}{5}\sqrt{3}\times \frac{\sqrt{3}}{4}\times \frac{1}{\sqrt{2}}$$

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

$$\frac{4}{5}\sqrt{3}\times \frac{\sqrt{3}}{4}\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{4}{5}\sqrt{3}\times \frac{\sqrt{3}}{4}\times \frac{\sqrt{2}}{2}$$

Multiply \frac{4}{5} times \frac{\sqrt{3}}{4} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\sqrt{3}}{5\times 4}\sqrt{3}\times \frac{\sqrt{2}}{2}$$

Cancel out 4 in both numerator and denominator.

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

Multiply \frac{\sqrt{3}}{5} times \frac{\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

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

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

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

Multiply 5 and 2 to get 10.

$$\frac{\sqrt{6}}{10}\sqrt{3}$$

Express \frac{\sqrt{6}}{10}\sqrt{3} as a single fraction.

$$\frac{\sqrt{6}\sqrt{3}}{10}$$

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

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

Multiply \sqrt{3} and \sqrt{3} to get 3.

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