Solve (4sqrt{2}-3sqrt{6})(2sqrt{6}+3sqrt{2})

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

Evaluate

$$-2\sqrt{3}-12\approx -15.464101615$$

Solution Steps

Apply the distributive property by multiplying each term of 4\sqrt{2}-3\sqrt{6} by each term of 2\sqrt{6}+3\sqrt{2}.

$$8\sqrt{2}\sqrt{6}+12\left(\sqrt{2}\right)^{2}-6\left(\sqrt{6}\right)^{2}-9\sqrt{6}\sqrt{2}$$

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}.

$$8\sqrt{2}\sqrt{2}\sqrt{3}+12\left(\sqrt{2}\right)^{2}-6\left(\sqrt{6}\right)^{2}-9\sqrt{6}\sqrt{2}$$

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

$$8\times 2\sqrt{3}+12\left(\sqrt{2}\right)^{2}-6\left(\sqrt{6}\right)^{2}-9\sqrt{6}\sqrt{2}$$

Multiply 8 and 2 to get 16.

$$16\sqrt{3}+12\left(\sqrt{2}\right)^{2}-6\left(\sqrt{6}\right)^{2}-9\sqrt{6}\sqrt{2}$$

The square of \sqrt{2} is 2.

$$16\sqrt{3}+12\times 2-6\left(\sqrt{6}\right)^{2}-9\sqrt{6}\sqrt{2}$$

Multiply 12 and 2 to get 24.

$$16\sqrt{3}+24-6\left(\sqrt{6}\right)^{2}-9\sqrt{6}\sqrt{2}$$

The square of \sqrt{6} is 6.

$$16\sqrt{3}+24-6\times 6-9\sqrt{6}\sqrt{2}$$

Multiply -6 and 6 to get -36.

$$16\sqrt{3}+24-36-9\sqrt{6}\sqrt{2}$$

Subtract 36 from 24 to get -12.

$$16\sqrt{3}-12-9\sqrt{6}\sqrt{2}$$

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}.

$$16\sqrt{3}-12-9\sqrt{2}\sqrt{3}\sqrt{2}$$

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

$$16\sqrt{3}-12-9\times 2\sqrt{3}$$

Multiply -9 and 2 to get -18.

$$16\sqrt{3}-12-18\sqrt{3}$$

Combine 16\sqrt{3} and -18\sqrt{3} to get -2\sqrt{3}.

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

Factor

$$2 {(-\sqrt{3} - 6)} = -15.464101615$$