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

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

Evaluate

$$\sqrt{2}\approx 1.414213562$$

Solution Steps

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

$$3\sqrt{2}-5\left(\sqrt{2}\right)^{2}+6-10\sqrt{2}-2\left(3\sqrt{2}-2\right)\left(-\sqrt{2}-2\right)$$

The square of \sqrt{2} is 2.

$$3\sqrt{2}-5\times 2+6-10\sqrt{2}-2\left(3\sqrt{2}-2\right)\left(-\sqrt{2}-2\right)$$

Multiply -5 and 2 to get -10.

$$3\sqrt{2}-10+6-10\sqrt{2}-2\left(3\sqrt{2}-2\right)\left(-\sqrt{2}-2\right)$$

Add -10 and 6 to get -4.

$$3\sqrt{2}-4-10\sqrt{2}-2\left(3\sqrt{2}-2\right)\left(-\sqrt{2}-2\right)$$

Combine 3\sqrt{2} and -10\sqrt{2} to get -7\sqrt{2}.

$$-7\sqrt{2}-4-2\left(3\sqrt{2}-2\right)\left(-\sqrt{2}-2\right)$$

Use the distributive property to multiply -2 by 3\sqrt{2}-2.

$$-7\sqrt{2}-4+\left(-6\sqrt{2}+4\right)\left(-\sqrt{2}-2\right)$$

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

$$-7\sqrt{2}-4+6\left(\sqrt{2}\right)^{2}+12\sqrt{2}-4\sqrt{2}-8$$

The square of \sqrt{2} is 2.

$$-7\sqrt{2}-4+6\times 2+12\sqrt{2}-4\sqrt{2}-8$$

Multiply 6 and 2 to get 12.

$$-7\sqrt{2}-4+12+12\sqrt{2}-4\sqrt{2}-8$$

Combine 12\sqrt{2} and -4\sqrt{2} to get 8\sqrt{2}.

$$-7\sqrt{2}-4+12+8\sqrt{2}-8$$

Subtract 8 from 12 to get 4.

$$-7\sqrt{2}-4+4+8\sqrt{2}$$

Add -4 and 4 to get 0.

$$-7\sqrt{2}+8\sqrt{2}$$

Combine -7\sqrt{2} and 8\sqrt{2} to get \sqrt{2}.

$$\sqrt{2}$$