Solve (2sqrt{6}+sqrt{12})(sqrt{96}-3sqrt{3})-2sqrt{72}

$$( 2 \sqrt { 6 } + \sqrt { 12 } ) ( \sqrt { 96 } - 3 \sqrt { 3 } ) - 2 \sqrt { 72 }$$

Evaluate

$$30-6\sqrt{2}\approx 21.514718626$$

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}.

$$\left(2\sqrt{6}+2\sqrt{3}\right)\left(\sqrt{96}-3\sqrt{3}\right)-2\sqrt{72}$$

Factor 96=4^{2}\times 6. Rewrite the square root of the product \sqrt{4^{2}\times 6} as the product of square roots \sqrt{4^{2}}\sqrt{6}. Take the square root of 4^{2}.

$$\left(2\sqrt{6}+2\sqrt{3}\right)\left(4\sqrt{6}-3\sqrt{3}\right)-2\sqrt{72}$$

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

$$8\left(\sqrt{6}\right)^{2}-6\sqrt{3}\sqrt{6}+8\sqrt{3}\sqrt{6}-6\left(\sqrt{3}\right)^{2}-2\sqrt{72}$$

The square of \sqrt{6} is 6.

$$8\times 6-6\sqrt{3}\sqrt{6}+8\sqrt{3}\sqrt{6}-6\left(\sqrt{3}\right)^{2}-2\sqrt{72}$$

Multiply 8 and 6 to get 48.

$$48-6\sqrt{3}\sqrt{6}+8\sqrt{3}\sqrt{6}-6\left(\sqrt{3}\right)^{2}-2\sqrt{72}$$

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}.

$$48-6\sqrt{3}\sqrt{3}\sqrt{2}+8\sqrt{3}\sqrt{6}-6\left(\sqrt{3}\right)^{2}-2\sqrt{72}$$

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

$$48-6\times 3\sqrt{2}+8\sqrt{3}\sqrt{6}-6\left(\sqrt{3}\right)^{2}-2\sqrt{72}$$

Multiply -6 and 3 to get -18.

$$48-18\sqrt{2}+8\sqrt{3}\sqrt{6}-6\left(\sqrt{3}\right)^{2}-2\sqrt{72}$$

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}.

$$48-18\sqrt{2}+8\sqrt{3}\sqrt{3}\sqrt{2}-6\left(\sqrt{3}\right)^{2}-2\sqrt{72}$$

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

$$48-18\sqrt{2}+8\times 3\sqrt{2}-6\left(\sqrt{3}\right)^{2}-2\sqrt{72}$$

Multiply 8 and 3 to get 24.

$$48-18\sqrt{2}+24\sqrt{2}-6\left(\sqrt{3}\right)^{2}-2\sqrt{72}$$

Combine -18\sqrt{2} and 24\sqrt{2} to get 6\sqrt{2}.

$$48+6\sqrt{2}-6\left(\sqrt{3}\right)^{2}-2\sqrt{72}$$

The square of \sqrt{3} is 3.

$$48+6\sqrt{2}-6\times 3-2\sqrt{72}$$

Multiply -6 and 3 to get -18.

$$48+6\sqrt{2}-18-2\sqrt{72}$$

Subtract 18 from 48 to get 30.

$$30+6\sqrt{2}-2\sqrt{72}$$

Factor 72=6^{2}\times 2. Rewrite the square root of the product \sqrt{6^{2}\times 2} as the product of square roots \sqrt{6^{2}}\sqrt{2}. Take the square root of 6^{2}.

$$30+6\sqrt{2}-2\times 6\sqrt{2}$$

Multiply -2 and 6 to get -12.

$$30+6\sqrt{2}-12\sqrt{2}$$

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

$$30-6\sqrt{2}$$