Solve 2sqrt{6}(sqrt{18}-24sqrt{11/3})-2sqrt{3}(sqrt{24}+4)

$$2 \sqrt { 6 } ( \sqrt { 18 } - 24 \sqrt { 1 \frac { 1 } { 3 } } ) - 2 \sqrt { 3 } ( \sqrt { 24 } + 4 )$$

Evaluate

$$4\left(\sqrt{3}-27\sqrt{2}\right)\approx -145.806861506$$

Solution Steps

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$2\sqrt{6}\left(3\sqrt{2}-24\sqrt{\frac{1\times 3+1}{3}}\right)-2\sqrt{3}\left(\sqrt{24}+4\right)$$

Multiply 1 and 3 to get 3.

$$2\sqrt{6}\left(3\sqrt{2}-24\sqrt{\frac{3+1}{3}}\right)-2\sqrt{3}\left(\sqrt{24}+4\right)$$

Add 3 and 1 to get 4.

$$2\sqrt{6}\left(3\sqrt{2}-24\sqrt{\frac{4}{3}}\right)-2\sqrt{3}\left(\sqrt{24}+4\right)$$

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

$$2\sqrt{6}\left(3\sqrt{2}-24\times \frac{\sqrt{4}}{\sqrt{3}}\right)-2\sqrt{3}\left(\sqrt{24}+4\right)$$

Calculate the square root of 4 and get 2.

$$2\sqrt{6}\left(3\sqrt{2}-24\times \frac{2}{\sqrt{3}}\right)-2\sqrt{3}\left(\sqrt{24}+4\right)$$

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

$$2\sqrt{6}\left(3\sqrt{2}-24\times \frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)-2\sqrt{3}\left(\sqrt{24}+4\right)$$

The square of \sqrt{3} is 3.

$$2\sqrt{6}\left(3\sqrt{2}-24\times \frac{2\sqrt{3}}{3}\right)-2\sqrt{3}\left(\sqrt{24}+4\right)$$

Cancel out 3, the greatest common factor in 24 and 3.

$$2\sqrt{6}\left(3\sqrt{2}-8\times 2\sqrt{3}\right)-2\sqrt{3}\left(\sqrt{24}+4\right)$$

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}.

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

Multiply 8 and 2 to get 16.

$$2\sqrt{6}\left(3\sqrt{2}-16\sqrt{3}\right)-2\sqrt{3}\left(2\sqrt{6}+4\right)$$

Use the distributive property to multiply 2\sqrt{6} by 3\sqrt{2}-16\sqrt{3}.

$$6\sqrt{6}\sqrt{2}-32\sqrt{3}\sqrt{6}-2\sqrt{3}\left(2\sqrt{6}+4\right)$$

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}.

$$6\sqrt{2}\sqrt{3}\sqrt{2}-32\sqrt{3}\sqrt{6}-2\sqrt{3}\left(2\sqrt{6}+4\right)$$

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

$$6\times 2\sqrt{3}-32\sqrt{3}\sqrt{6}-2\sqrt{3}\left(2\sqrt{6}+4\right)$$

Multiply 6 and 2 to get 12.

$$12\sqrt{3}-32\sqrt{3}\sqrt{6}-2\sqrt{3}\left(2\sqrt{6}+4\right)$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$12\sqrt{3}-32\sqrt{3}\sqrt{3}\sqrt{2}-2\sqrt{3}\left(2\sqrt{6}+4\right)$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$12\sqrt{3}-32\times 3\sqrt{2}-2\sqrt{3}\left(2\sqrt{6}+4\right)$$

Multiply -32 and 3 to get -96.

$$12\sqrt{3}-96\sqrt{2}-2\sqrt{3}\left(2\sqrt{6}+4\right)$$

Use the distributive property to multiply -2\sqrt{3} by 2\sqrt{6}+4.

$$12\sqrt{3}-96\sqrt{2}-4\sqrt{3}\sqrt{6}-8\sqrt{3}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$12\sqrt{3}-96\sqrt{2}-4\sqrt{3}\sqrt{3}\sqrt{2}-8\sqrt{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$12\sqrt{3}-96\sqrt{2}-4\times 3\sqrt{2}-8\sqrt{3}$$

Multiply -4 and 3 to get -12.

$$12\sqrt{3}-96\sqrt{2}-12\sqrt{2}-8\sqrt{3}$$

Combine -96\sqrt{2} and -12\sqrt{2} to get -108\sqrt{2}.

$$12\sqrt{3}-108\sqrt{2}-8\sqrt{3}$$

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

$$4\sqrt{3}-108\sqrt{2}$$

Factor

$$4 {(\sqrt{3} - 27 \sqrt{2})} = -145.806861506$$