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

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

Evaluate

$$-\frac{145\sqrt{3}}{3}+14\approx -69.715789032$$

Short Solution Steps

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

$$\left(\frac{\sqrt{1}}{\sqrt{2}}-6\sqrt{\frac{3}{2}}\right)\left(4\sqrt{8}-\sqrt{\frac{2}{3}}\right)$$

Calculate the square root of 1 and get 1.

$$\left(\frac{1}{\sqrt{2}}-6\sqrt{\frac{3}{2}}\right)\left(4\sqrt{8}-\sqrt{\frac{2}{3}}\right)$$

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

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

The square of \sqrt{2} is 2.

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

Rewrite the square root of the division \sqrt{\frac{3}{2}} as the division of square roots \frac{\sqrt{3}}{\sqrt{2}}.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Cancel out 2, the greatest common factor in 6 and 2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -3\sqrt{6} times \frac{2}{2}.

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

Since \frac{\sqrt{2}}{2} and \frac{2\left(-3\right)\sqrt{6}}{2} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \sqrt{2}+2\left(-3\right)\sqrt{6}.

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

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}.

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

Multiply 4 and 2 to get 8.

$$\frac{\sqrt{2}-6\sqrt{6}}{2}\left(8\sqrt{2}-\sqrt{\frac{2}{3}}\right)$$

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

$$\frac{\sqrt{2}-6\sqrt{6}}{2}\left(8\sqrt{2}-\frac{\sqrt{2}}{\sqrt{3}}\right)$$

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

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

The square of \sqrt{3} is 3.

$$\frac{\sqrt{2}-6\sqrt{6}}{2}\left(8\sqrt{2}-\frac{\sqrt{2}\sqrt{3}}{3}\right)$$

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

$$\frac{\sqrt{2}-6\sqrt{6}}{2}\left(8\sqrt{2}-\frac{\sqrt{6}}{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 8\sqrt{2} times \frac{3}{3}.

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

Since \frac{3\times 8\sqrt{2}}{3} and \frac{\sqrt{6}}{3} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\times 8\sqrt{2}-\sqrt{6}.

$$\frac{\sqrt{2}-6\sqrt{6}}{2}\times \frac{24\sqrt{2}-\sqrt{6}}{3}$$

Multiply \frac{\sqrt{2}-6\sqrt{6}}{2} times \frac{24\sqrt{2}-\sqrt{6}}{3} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(\sqrt{2}-6\sqrt{6}\right)\left(24\sqrt{2}-\sqrt{6}\right)}{2\times 3}$$

Multiply 2 and 3 to get 6.

$$\frac{\left(\sqrt{2}-6\sqrt{6}\right)\left(24\sqrt{2}-\sqrt{6}\right)}{6}$$

Apply the distributive property by multiplying each term of \sqrt{2}-6\sqrt{6} by each term of 24\sqrt{2}-\sqrt{6}.

$$\frac{24\left(\sqrt{2}\right)^{2}-\sqrt{2}\sqrt{6}-144\sqrt{6}\sqrt{2}+6\left(\sqrt{6}\right)^{2}}{6}$$

The square of \sqrt{2} is 2.

$$\frac{24\times 2-\sqrt{2}\sqrt{6}-144\sqrt{6}\sqrt{2}+6\left(\sqrt{6}\right)^{2}}{6}$$

Multiply 24 and 2 to get 48.

$$\frac{48-\sqrt{2}\sqrt{6}-144\sqrt{6}\sqrt{2}+6\left(\sqrt{6}\right)^{2}}{6}$$

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}.

$$\frac{48-\sqrt{2}\sqrt{2}\sqrt{3}-144\sqrt{6}\sqrt{2}+6\left(\sqrt{6}\right)^{2}}{6}$$

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

$$\frac{48-2\sqrt{3}-144\sqrt{6}\sqrt{2}+6\left(\sqrt{6}\right)^{2}}{6}$$

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}.

$$\frac{48-2\sqrt{3}-144\sqrt{2}\sqrt{3}\sqrt{2}+6\left(\sqrt{6}\right)^{2}}{6}$$

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

$$\frac{48-2\sqrt{3}-144\times 2\sqrt{3}+6\left(\sqrt{6}\right)^{2}}{6}$$

Multiply -144 and 2 to get -288.

$$\frac{48-2\sqrt{3}-288\sqrt{3}+6\left(\sqrt{6}\right)^{2}}{6}$$

Combine -2\sqrt{3} and -288\sqrt{3} to get -290\sqrt{3}.

$$\frac{48-290\sqrt{3}+6\left(\sqrt{6}\right)^{2}}{6}$$

The square of \sqrt{6} is 6.

$$\frac{48-290\sqrt{3}+6\times 6}{6}$$

Multiply 6 and 6 to get 36.

$$\frac{48-290\sqrt{3}+36}{6}$$

Add 48 and 36 to get 84.

$$\frac{84-290\sqrt{3}}{6}$$