Solve (sqrt{3}+sqrt{6}+sqrt{12})*(sqrt{3}-sqrt{6}-sqrt{12})

$$( \sqrt { 3 } + \sqrt { 6 } + \sqrt { 12 } ) \times ( \sqrt { 3 } - \sqrt { 6 } - \sqrt { 12 } )$$

Evaluate

$$-12\sqrt{2}-15\approx -31.970562748$$

Solution Steps

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}.

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

Combine \sqrt{3} and 2\sqrt{3} to get 3\sqrt{3}.

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

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}.

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Multiply -3 and 3 to get -9.

$$-9-3\sqrt{3}\sqrt{6}-\sqrt{6}\sqrt{3}-\left(\sqrt{6}\right)^{2}$$

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}.

$$-9-3\sqrt{3}\sqrt{3}\sqrt{2}-\sqrt{6}\sqrt{3}-\left(\sqrt{6}\right)^{2}$$

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

$$-9-3\times 3\sqrt{2}-\sqrt{6}\sqrt{3}-\left(\sqrt{6}\right)^{2}$$

Multiply -3 and 3 to get -9.

$$-9-9\sqrt{2}-\sqrt{6}\sqrt{3}-\left(\sqrt{6}\right)^{2}$$

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}.

$$-9-9\sqrt{2}-\sqrt{3}\sqrt{2}\sqrt{3}-\left(\sqrt{6}\right)^{2}$$

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

$$-9-9\sqrt{2}-3\sqrt{2}-\left(\sqrt{6}\right)^{2}$$

Combine -9\sqrt{2} and -3\sqrt{2} to get -12\sqrt{2}.

$$-9-12\sqrt{2}-\left(\sqrt{6}\right)^{2}$$

The square of \sqrt{6} is 6.

$$-9-12\sqrt{2}-6$$

Subtract 6 from -9 to get -15.

$$-15-12\sqrt{2}$$