Solve (4sqrt{6}-4sqrt{1/2}+3sqrt{8})div2sqrt{2}=

$$( 4 \sqrt { 6 } - 4 \sqrt { \frac { 1 } { 2 } } + 3 \sqrt { 8 } ) \div 2 \sqrt { 2 } =$$

Evaluate

$$4\left(\sqrt{3}+1\right)\approx 10.92820323$$

Solution Steps

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

$$\frac{4\sqrt{6}-4\times \frac{\sqrt{1}}{\sqrt{2}}+3\sqrt{8}}{2}\sqrt{2}$$

Calculate the square root of 1 and get 1.

$$\frac{4\sqrt{6}-4\times \frac{1}{\sqrt{2}}+3\sqrt{8}}{2}\sqrt{2}$$

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

$$\frac{4\sqrt{6}-4\times \frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+3\sqrt{8}}{2}\sqrt{2}$$

The square of \sqrt{2} is 2.

$$\frac{4\sqrt{6}-4\times \frac{\sqrt{2}}{2}+3\sqrt{8}}{2}\sqrt{2}$$

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

$$\frac{4\sqrt{6}-2\sqrt{2}+3\sqrt{8}}{2}\sqrt{2}$$

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}.

$$\frac{4\sqrt{6}-2\sqrt{2}+3\times 2\sqrt{2}}{2}\sqrt{2}$$

Multiply 3 and 2 to get 6.

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

Combine -2\sqrt{2} and 6\sqrt{2} to get 4\sqrt{2}.

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

Express \frac{4\sqrt{6}+4\sqrt{2}}{2}\sqrt{2} as a single fraction.

$$\frac{\left(4\sqrt{6}+4\sqrt{2}\right)\sqrt{2}}{2}$$

Use the distributive property to multiply 4\sqrt{6}+4\sqrt{2} by \sqrt{2}.

$$\frac{4\sqrt{6}\sqrt{2}+4\left(\sqrt{2}\right)^{2}}{2}$$

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}.

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

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

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

Multiply 4 and 2 to get 8.

$$\frac{8\sqrt{3}+4\left(\sqrt{2}\right)^{2}}{2}$$

The square of \sqrt{2} is 2.

$$\frac{8\sqrt{3}+4\times 2}{2}$$

Multiply 4 and 2 to get 8.

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