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

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

Evaluate

$$4-9\sqrt{6}\approx -18.045407685$$

Solution Steps

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

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

The square of \sqrt{2} is 2.

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

Multiply 5 and 2 to get 10.

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

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

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

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

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

Combine -10\sqrt{6} and \sqrt{6} to get -9\sqrt{6}.

$$10-9\sqrt{6}-2\left(\sqrt{3}\right)^{2}$$

The square of \sqrt{3} is 3.

$$10-9\sqrt{6}-2\times 3$$

Multiply -2 and 3 to get -6.

$$10-9\sqrt{6}-6$$

Subtract 6 from 10 to get 4.

$$4-9\sqrt{6}$$