Solve (2sqrt{8}+3sqrt{5}-7sqrt{2})*(sqrt{72}+sqrt{20}-4sqrt{2})

$$( 2 \sqrt { 8 } + 3 \sqrt { 5 } - 7 \sqrt { 2 } ) \cdot ( \sqrt { 72 } + \sqrt { 20 } - 4 \sqrt { 2 } )$$

Evaluate

$$18$$

Solution Steps

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}.

$$\left(2\times 2\sqrt{2}+3\sqrt{5}-7\sqrt{2}\right)\left(\sqrt{72}+\sqrt{20}-4\sqrt{2}\right)$$

Multiply 2 and 2 to get 4.

$$\left(4\sqrt{2}+3\sqrt{5}-7\sqrt{2}\right)\left(\sqrt{72}+\sqrt{20}-4\sqrt{2}\right)$$

Combine 4\sqrt{2} and -7\sqrt{2} to get -3\sqrt{2}.

$$\left(-3\sqrt{2}+3\sqrt{5}\right)\left(\sqrt{72}+\sqrt{20}-4\sqrt{2}\right)$$

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}.

$$\left(-3\sqrt{2}+3\sqrt{5}\right)\left(6\sqrt{2}+\sqrt{20}-4\sqrt{2}\right)$$

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\left(-3\sqrt{2}+3\sqrt{5}\right)\left(6\sqrt{2}+2\sqrt{5}-4\sqrt{2}\right)$$

Combine 6\sqrt{2} and -4\sqrt{2} to get 2\sqrt{2}.

$$\left(-3\sqrt{2}+3\sqrt{5}\right)\left(2\sqrt{2}+2\sqrt{5}\right)$$

Apply the distributive property by multiplying each term of -3\sqrt{2}+3\sqrt{5} by each term of 2\sqrt{2}+2\sqrt{5}.

$$-6\left(\sqrt{2}\right)^{2}-6\sqrt{2}\sqrt{5}+6\sqrt{5}\sqrt{2}+6\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{2} is 2.

$$-6\times 2-6\sqrt{2}\sqrt{5}+6\sqrt{5}\sqrt{2}+6\left(\sqrt{5}\right)^{2}$$

Multiply -6 and 2 to get -12.

$$-12-6\sqrt{2}\sqrt{5}+6\sqrt{5}\sqrt{2}+6\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$-12-6\sqrt{10}+6\sqrt{5}\sqrt{2}+6\left(\sqrt{5}\right)^{2}$$

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$-12-6\sqrt{10}+6\sqrt{10}+6\left(\sqrt{5}\right)^{2}$$

Combine -6\sqrt{10} and 6\sqrt{10} to get 0.

$$-12+6\left(\sqrt{5}\right)^{2}$$

The square of \sqrt{5} is 5.

$$-12+6\times 5$$

Multiply 6 and 5 to get 30.

$$-12+30$$

Add -12 and 30 to get 18.

$$18$$

Factor

$$2\times 3^{2}$$