Solve (1+5sqrt{2})(1-2sqrt{2})-sqrt{18}

$$(1+5 \sqrt{ 2 } )(1-2 \sqrt{ 2 } )- \sqrt{ 18 }$$

Evaluate

$$-19$$

Solution Steps

Apply the distributive property by multiplying each term of 1+5\sqrt{2} by each term of 1-2\sqrt{2}.

$$1-2\sqrt{2}+5\sqrt{2}-10\left(\sqrt{2}\right)^{2}-\sqrt{18}$$

Combine -2\sqrt{2} and 5\sqrt{2} to get 3\sqrt{2}.

$$1+3\sqrt{2}-10\left(\sqrt{2}\right)^{2}-\sqrt{18}$$

The square of \sqrt{2} is 2.

$$1+3\sqrt{2}-10\times 2-\sqrt{18}$$

Multiply -10 and 2 to get -20.

$$1+3\sqrt{2}-20-\sqrt{18}$$

Subtract 20 from 1 to get -19.

$$-19+3\sqrt{2}-\sqrt{18}$$

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

$$-19+3\sqrt{2}-3\sqrt{2}$$

Combine 3\sqrt{2} and -3\sqrt{2} to get 0.

$$-19$$

Factor

$$-19$$