Solve frac{3sqrt{2}1/2sqrt{6}}{sqrt{8}}

$$\frac{ 3 \sqrt{ 2 } \frac{ 1 }{ 2 } \sqrt{ 6 } }{ \sqrt{ 8 } }$$

Evaluate

$$\frac{3\sqrt{6}}{4}\approx 1.837117307$$

Solution Steps

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}.

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

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

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

Multiply 3 and 2 to get 6.

$$\frac{6\times \frac{1}{2}\sqrt{3}}{\sqrt{8}}$$

Multiply 6 and \frac{1}{2} to get \frac{6}{2}.

$$\frac{\frac{6}{2}\sqrt{3}}{\sqrt{8}}$$

Divide 6 by 2 to get 3.

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

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}.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Multiply 2 and 2 to get 4.

$$\frac{3\sqrt{6}}{4}$$