Solve (6+sqrt{3}i)(2-sqrt{6}i)

$$( 6 + \sqrt { 3 } i ) ( 2 - \sqrt { 6 } i )$$

Evaluate

$$-6\sqrt{6}i+3\sqrt{2}+12+2\sqrt{3}i\approx 16.242640687-11.232836842i$$

Solution Steps

Use the distributive property to multiply 6+i\sqrt{3} by 2-i\sqrt{6}.

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

Use the distributive property to multiply 6 by 2-i\sqrt{6}.

$$12-6i\sqrt{6}+i\sqrt{3}\left(2-i\sqrt{6}\right)$$

Use the distributive property to multiply i\sqrt{3} by 2-i\sqrt{6}.

$$12-6i\sqrt{6}+2i\sqrt{3}+\sqrt{3}\sqrt{6}$$

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}.

$$12-6i\sqrt{6}+2i\sqrt{3}+\sqrt{3}\sqrt{3}\sqrt{2}$$

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

$$12-6i\sqrt{6}+2i\sqrt{3}+3\sqrt{2}$$

Real Part

$$3 {(\sqrt{2} + 4)} = 16.242640687$$