Solve (sqrt{12}-3sqrt{2})(4sqrt{3}+sqrt{8})-2sqrt{24}

$$( \sqrt{ 12 } -3 \sqrt{ 2 } )(4 \sqrt{ 3 } + \sqrt{ 8 } )-2 \sqrt{ 24 }$$

Evaluate

$$12-12\sqrt{6}\approx -17.393876913$$

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}.

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

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}.

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

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

$$8\left(\sqrt{3}\right)^{2}+4\sqrt{3}\sqrt{2}-12\sqrt{3}\sqrt{2}-6\left(\sqrt{2}\right)^{2}-2\sqrt{24}$$

The square of \sqrt{3} is 3.

$$8\times 3+4\sqrt{3}\sqrt{2}-12\sqrt{3}\sqrt{2}-6\left(\sqrt{2}\right)^{2}-2\sqrt{24}$$

Multiply 8 and 3 to get 24.

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

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

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

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

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

Combine 4\sqrt{6} and -12\sqrt{6} to get -8\sqrt{6}.

$$24-8\sqrt{6}-6\left(\sqrt{2}\right)^{2}-2\sqrt{24}$$

The square of \sqrt{2} is 2.

$$24-8\sqrt{6}-6\times 2-2\sqrt{24}$$

Multiply -6 and 2 to get -12.

$$24-8\sqrt{6}-12-2\sqrt{24}$$

Subtract 12 from 24 to get 12.

$$12-8\sqrt{6}-2\sqrt{24}$$

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}.

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

Multiply -2 and 2 to get -4.

$$12-8\sqrt{6}-4\sqrt{6}$$

Combine -8\sqrt{6} and -4\sqrt{6} to get -12\sqrt{6}.

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