Solve sqrt{48}*sqrt{1/2}div3sqrt{2}

$$\sqrt { 48 } \times \sqrt { \frac { 1 } { 2 } } \div 3 \sqrt { 2 }$$

Evaluate

$$\frac{4\sqrt{3}}{3}\approx 2.309401077$$

Solution Steps

Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

Cancel out 2, the greatest common factor in 4 and 2.

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

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

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

Express \frac{2\sqrt{6}}{3}\sqrt{2} as a single fraction.

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

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}.

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

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

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

Multiply 2 and 2 to get 4.

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