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

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

Evaluate

$$4\left(\sqrt{2}+\sqrt{6}\right)\approx 15.454813221$$

Solution Steps

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

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}.

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

Multiply 3 and 2 to get 6.

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

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

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