Solve (5-2sqrt{6})(sqrt{2}-sqrt{3})

$$( 5 - 2 \sqrt { 6 } ) ( \sqrt { 2 } - \sqrt { 3 } )$$

Evaluate

$$11\sqrt{2}-9\sqrt{3}\approx -0.032108082$$

Solution Steps

Apply the distributive property by multiplying each term of 5-2\sqrt{6} by each term of \sqrt{2}-\sqrt{3}.

$$5\sqrt{2}-5\sqrt{3}-2\sqrt{6}\sqrt{2}+2\sqrt{3}\sqrt{6}$$

Factor 6=2\times 3. Rewrite the square root of the product \sqrt{2\times 3} as the product of square roots \sqrt{2}\sqrt{3}.

$$5\sqrt{2}-5\sqrt{3}-2\sqrt{2}\sqrt{3}\sqrt{2}+2\sqrt{3}\sqrt{6}$$

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

$$5\sqrt{2}-5\sqrt{3}-2\times 2\sqrt{3}+2\sqrt{3}\sqrt{6}$$

Multiply -2 and 2 to get -4.

$$5\sqrt{2}-5\sqrt{3}-4\sqrt{3}+2\sqrt{3}\sqrt{6}$$

Combine -5\sqrt{3} and -4\sqrt{3} to get -9\sqrt{3}.

$$5\sqrt{2}-9\sqrt{3}+2\sqrt{3}\sqrt{6}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$5\sqrt{2}-9\sqrt{3}+2\sqrt{3}\sqrt{3}\sqrt{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$5\sqrt{2}-9\sqrt{3}+2\times 3\sqrt{2}$$

Multiply 2 and 3 to get 6.

$$5\sqrt{2}-9\sqrt{3}+6\sqrt{2}$$

Combine 5\sqrt{2} and 6\sqrt{2} to get 11\sqrt{2}.

$$11\sqrt{2}-9\sqrt{3}$$