Solve (5sqrt{2}-2sqrt{10})(-3sqrt{10}-2sqrt{2})

$$( 5 \sqrt { 2 } - 2 \sqrt { 10 } ) ( - 3 \sqrt { 10 } - 2 \sqrt { 2 } )$$

Evaluate

$$40-22\sqrt{5}\approx -9.193495505$$

Solution Steps

Apply the distributive property by multiplying each term of 5\sqrt{2}-2\sqrt{10} by each term of -3\sqrt{10}-2\sqrt{2}.

$$-15\sqrt{10}\sqrt{2}-10\left(\sqrt{2}\right)^{2}+6\left(\sqrt{10}\right)^{2}+4\sqrt{2}\sqrt{10}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$-15\sqrt{2}\sqrt{5}\sqrt{2}-10\left(\sqrt{2}\right)^{2}+6\left(\sqrt{10}\right)^{2}+4\sqrt{2}\sqrt{10}$$

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

$$-15\times 2\sqrt{5}-10\left(\sqrt{2}\right)^{2}+6\left(\sqrt{10}\right)^{2}+4\sqrt{2}\sqrt{10}$$

Multiply -15 and 2 to get -30.

$$-30\sqrt{5}-10\left(\sqrt{2}\right)^{2}+6\left(\sqrt{10}\right)^{2}+4\sqrt{2}\sqrt{10}$$

The square of \sqrt{2} is 2.

$$-30\sqrt{5}-10\times 2+6\left(\sqrt{10}\right)^{2}+4\sqrt{2}\sqrt{10}$$

Multiply -10 and 2 to get -20.

$$-30\sqrt{5}-20+6\left(\sqrt{10}\right)^{2}+4\sqrt{2}\sqrt{10}$$

The square of \sqrt{10} is 10.

$$-30\sqrt{5}-20+6\times 10+4\sqrt{2}\sqrt{10}$$

Multiply 6 and 10 to get 60.

$$-30\sqrt{5}-20+60+4\sqrt{2}\sqrt{10}$$

Add -20 and 60 to get 40.

$$-30\sqrt{5}+40+4\sqrt{2}\sqrt{10}$$

Factor 10=2\times 5. Rewrite the square root of the product \sqrt{2\times 5} as the product of square roots \sqrt{2}\sqrt{5}.

$$-30\sqrt{5}+40+4\sqrt{2}\sqrt{2}\sqrt{5}$$

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

$$-30\sqrt{5}+40+4\times 2\sqrt{5}$$

Multiply 4 and 2 to get 8.

$$-30\sqrt{5}+40+8\sqrt{5}$$

Combine -30\sqrt{5} and 8\sqrt{5} to get -22\sqrt{5}.

$$-22\sqrt{5}+40$$