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

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

Evaluate

$$17-5\sqrt{10}\approx 1.188611699$$

Solution Steps

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}.

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

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

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

The square of \sqrt{5} is 5.

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

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

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

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

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

Combine -2\sqrt{10} and -3\sqrt{10} to get -5\sqrt{10}.

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

The square of \sqrt{2} is 2.

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

Multiply 6 and 2 to get 12.

$$5-5\sqrt{10}+12$$

Add 5 and 12 to get 17.

$$17-5\sqrt{10}$$