Solve left(2sqrt{6}-sqrt{3}+sqrt{2}right)left(2sqrt{6}-sqrt{3}-sqrt{2}right)

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

Evaluate

$$25-12\sqrt{2}\approx 8.029437252$$

Solution Steps

Apply the distributive property by multiplying each term of 2\sqrt{6}-\sqrt{3}+\sqrt{2} by each term of 2\sqrt{6}-\sqrt{3}-\sqrt{2}.

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

The square of \sqrt{6} is 6.

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

Multiply 4 and 6 to get 24.

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

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}.

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

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

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

Multiply -2 and 3 to get -6.

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

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}.

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

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

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

Multiply -2 and 2 to get -4.

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

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}.

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

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

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

Multiply -2 and 3 to get -6.

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

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

$$24-12\sqrt{2}-4\sqrt{3}+\left(\sqrt{3}\right)^{2}+\sqrt{3}\sqrt{2}+2\sqrt{2}\sqrt{6}-\sqrt{2}\sqrt{3}-\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{3} is 3.

$$24-12\sqrt{2}-4\sqrt{3}+3+\sqrt{3}\sqrt{2}+2\sqrt{2}\sqrt{6}-\sqrt{2}\sqrt{3}-\left(\sqrt{2}\right)^{2}$$

Add 24 and 3 to get 27.

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

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

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

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}.

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

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

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

Multiply 2 and 2 to get 4.

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

Combine -4\sqrt{3} and 4\sqrt{3} to get 0.

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

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

$$27-12\sqrt{2}+\sqrt{6}-\sqrt{6}-\left(\sqrt{2}\right)^{2}$$

Combine \sqrt{6} and -\sqrt{6} to get 0.

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

The square of \sqrt{2} is 2.

$$27-12\sqrt{2}-2$$

Subtract 2 from 27 to get 25.

$$25-12\sqrt{2}$$