Solve (4)(3+sqrt{10})(sqrt{2}-sqrt{5})

$$( 4 ) ( 3 + \sqrt { 10 } ) ( \sqrt { 2 } - \sqrt { 5 } )$$

Evaluate

$$-4\sqrt{5}-8\sqrt{2}\approx -20.257980409$$

Solution Steps

Use the distributive property to multiply 4 by 3+\sqrt{10}.

$$\left(12+4\sqrt{10}\right)\left(\sqrt{2}-\sqrt{5}\right)$$

Apply the distributive property by multiplying each term of 12+4\sqrt{10} by each term of \sqrt{2}-\sqrt{5}.

$$12\sqrt{2}-12\sqrt{5}+4\sqrt{10}\sqrt{2}-4\sqrt{10}\sqrt{5}$$

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}.

$$12\sqrt{2}-12\sqrt{5}+4\sqrt{2}\sqrt{5}\sqrt{2}-4\sqrt{10}\sqrt{5}$$

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

$$12\sqrt{2}-12\sqrt{5}+4\times 2\sqrt{5}-4\sqrt{10}\sqrt{5}$$

Multiply 4 and 2 to get 8.

$$12\sqrt{2}-12\sqrt{5}+8\sqrt{5}-4\sqrt{10}\sqrt{5}$$

Combine -12\sqrt{5} and 8\sqrt{5} to get -4\sqrt{5}.

$$12\sqrt{2}-4\sqrt{5}-4\sqrt{10}\sqrt{5}$$

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

$$12\sqrt{2}-4\sqrt{5}-4\sqrt{5}\sqrt{2}\sqrt{5}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$12\sqrt{2}-4\sqrt{5}-4\times 5\sqrt{2}$$

Multiply -4 and 5 to get -20.

$$12\sqrt{2}-4\sqrt{5}-20\sqrt{2}$$

Combine 12\sqrt{2} and -20\sqrt{2} to get -8\sqrt{2}.

$$-8\sqrt{2}-4\sqrt{5}$$