Solve (2)frac{sqrt{2}}{2}(2sqrt{12}+4sqrt{1/8}-3sqrt{48})-4

$$( 2 ) \frac { \sqrt { 2 } } { 2 } ( 2 \sqrt { 12 } + 4 \sqrt { \frac { 1 } { 8 } } - 3 \sqrt { 48 } ) - 4$$

Evaluate

$$-8\sqrt{6}-2\approx -21.595917942$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$2\times \frac{\sqrt{2}}{2}\left(2\times 2\sqrt{3}+4\sqrt{\frac{1}{8}}-3\sqrt{48}\right)-4$$

Multiply 2 and 2 to get 4.

$$2\times \frac{\sqrt{2}}{2}\left(4\sqrt{3}+4\sqrt{\frac{1}{8}}-3\sqrt{48}\right)-4$$

Rewrite the square root of the division \sqrt{\frac{1}{8}} as the division of square roots \frac{\sqrt{1}}{\sqrt{8}}.

$$2\times \frac{\sqrt{2}}{2}\left(4\sqrt{3}+4\times \frac{\sqrt{1}}{\sqrt{8}}-3\sqrt{48}\right)-4$$

Calculate the square root of 1 and get 1.

$$2\times \frac{\sqrt{2}}{2}\left(4\sqrt{3}+4\times \frac{1}{\sqrt{8}}-3\sqrt{48}\right)-4$$

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}.

$$2\times \frac{\sqrt{2}}{2}\left(4\sqrt{3}+4\times \frac{1}{2\sqrt{2}}-3\sqrt{48}\right)-4$$

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

$$2\times \frac{\sqrt{2}}{2}\left(4\sqrt{3}+4\times \frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}-3\sqrt{48}\right)-4$$

The square of \sqrt{2} is 2.

$$2\times \frac{\sqrt{2}}{2}\left(4\sqrt{3}+4\times \frac{\sqrt{2}}{2\times 2}-3\sqrt{48}\right)-4$$

Multiply 2 and 2 to get 4.

$$2\times \frac{\sqrt{2}}{2}\left(4\sqrt{3}+4\times \frac{\sqrt{2}}{4}-3\sqrt{48}\right)-4$$

Cancel out 4 and 4.

$$2\times \frac{\sqrt{2}}{2}\left(4\sqrt{3}+\sqrt{2}-3\sqrt{48}\right)-4$$

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}.

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

Multiply -3 and 4 to get -12.

$$2\times \frac{\sqrt{2}}{2}\left(4\sqrt{3}+\sqrt{2}-12\sqrt{3}\right)-4$$

Combine 4\sqrt{3} and -12\sqrt{3} to get -8\sqrt{3}.

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

Cancel out 2 and 2.

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

Use the distributive property to multiply \sqrt{2} by -8\sqrt{3}+\sqrt{2}.

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

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

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

The square of \sqrt{2} is 2.

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

Subtract 4 from 2 to get -2.

$$-8\sqrt{6}-2$$