Solve (sqrt{24}+sqrt{24})(sqrt{1/8}-sqrt{6})

$$( \sqrt { 24 } + \sqrt { 24 } ) ( \sqrt { \frac { 1 } { 8 } } - \sqrt { 6 } )$$

Evaluate

$$2\left(\sqrt{3}-12\right)\approx -20.535898385$$

Short Solution Steps

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

$$\left(2\sqrt{6}+\sqrt{24}\right)\left(\sqrt{\frac{1}{8}}-\sqrt{6}\right)$$

Factor 24=2^{2}\times 6. Rewrite the square root of the product \sqrt{2^{2}\times 6} as the product of square roots \sqrt{2^{2}}\sqrt{6}. Take the square root of 2^{2}.

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

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

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

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

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

Calculate the square root of 1 and get 1.

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

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}.

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{6} times \frac{4}{4}.

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

Since \frac{\sqrt{2}}{4} and \frac{4\sqrt{6}}{4} have the same denominator, subtract them by subtracting their numerators.

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

Cancel out 4 and 4.

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

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

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

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}.

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

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

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

The square of \sqrt{6} is 6.

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

Multiply -4 and 6 to get -24.

$$2\sqrt{3}-24$$

Factor

$$2 {(\sqrt{3} - 12)} = -20.535898385$$