Solve 2sqrt{6}left(2sqrt{3}-sqrt{8}right)-sqrt{3/2}left(2sqrt{27}-sqrt{12}right)

$$2 \sqrt{ 6 } \left( 2 \sqrt{ 3 } - \sqrt{ 8 } \right) - \sqrt{ \frac{ 3 }{ 2 } } \left( 2 \sqrt{ 27 } - \sqrt{ 12 } \right)$$

Evaluate

$$2\sqrt{3}\left(\sqrt{6}-4\right)\approx -5.371125086$$

Solution Steps

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}.

$$2\sqrt{6}\left(2\sqrt{3}-2\sqrt{2}\right)-\sqrt{\frac{3}{2}}\left(2\sqrt{27}-\sqrt{12}\right)$$

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

$$4\sqrt{3}\sqrt{6}-4\sqrt{6}\sqrt{2}-\sqrt{\frac{3}{2}}\left(2\sqrt{27}-\sqrt{12}\right)$$

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}.

$$4\sqrt{3}\sqrt{3}\sqrt{2}-4\sqrt{6}\sqrt{2}-\sqrt{\frac{3}{2}}\left(2\sqrt{27}-\sqrt{12}\right)$$

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

$$4\times 3\sqrt{2}-4\sqrt{6}\sqrt{2}-\sqrt{\frac{3}{2}}\left(2\sqrt{27}-\sqrt{12}\right)$$

Multiply 4 and 3 to get 12.

$$12\sqrt{2}-4\sqrt{6}\sqrt{2}-\sqrt{\frac{3}{2}}\left(2\sqrt{27}-\sqrt{12}\right)$$

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}.

$$12\sqrt{2}-4\sqrt{2}\sqrt{3}\sqrt{2}-\sqrt{\frac{3}{2}}\left(2\sqrt{27}-\sqrt{12}\right)$$

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

$$12\sqrt{2}-4\times 2\sqrt{3}-\sqrt{\frac{3}{2}}\left(2\sqrt{27}-\sqrt{12}\right)$$

Multiply -4 and 2 to get -8.

$$12\sqrt{2}-8\sqrt{3}-\sqrt{\frac{3}{2}}\left(2\sqrt{27}-\sqrt{12}\right)$$

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

$$12\sqrt{2}-8\sqrt{3}-\frac{\sqrt{3}}{\sqrt{2}}\left(2\sqrt{27}-\sqrt{12}\right)$$

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

$$12\sqrt{2}-8\sqrt{3}-\frac{\sqrt{3}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\left(2\sqrt{27}-\sqrt{12}\right)$$

The square of \sqrt{2} is 2.

$$12\sqrt{2}-8\sqrt{3}-\frac{\sqrt{3}\sqrt{2}}{2}\left(2\sqrt{27}-\sqrt{12}\right)$$

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

$$12\sqrt{2}-8\sqrt{3}-\frac{\sqrt{6}}{2}\left(2\sqrt{27}-\sqrt{12}\right)$$

Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.

$$12\sqrt{2}-8\sqrt{3}-\frac{\sqrt{6}}{2}\left(2\times 3\sqrt{3}-\sqrt{12}\right)$$

Multiply 2 and 3 to get 6.

$$12\sqrt{2}-8\sqrt{3}-\frac{\sqrt{6}}{2}\left(6\sqrt{3}-\sqrt{12}\right)$$

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}.

$$12\sqrt{2}-8\sqrt{3}-\frac{\sqrt{6}}{2}\left(6\sqrt{3}-2\sqrt{3}\right)$$

Combine 6\sqrt{3} and -2\sqrt{3} to get 4\sqrt{3}.

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

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

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

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}.

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

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

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

Multiply 2 and 3 to get 6.

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

Combine 12\sqrt{2} and -6\sqrt{2} to get 6\sqrt{2}.

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