Solve (-sqrt{2}+sqrt{3}i)-[(sqrt{3}-sqrt{2})+(sqrt{3}+sqrt{2})i]+(-sqrt{2}i+sqrt{3})

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

Evaluate

$$-2\sqrt{2}i\approx -0-2.828427125i$$

Solution Steps

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

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

Combine \sqrt{3} and i\sqrt{3} to get \left(1+i\right)\sqrt{3}.

$$-\sqrt{2}+i\sqrt{3}-\left(\left(1+i\right)\sqrt{3}-\sqrt{2}+i\sqrt{2}\right)+i\left(-\sqrt{2}\right)+\sqrt{3}$$

Combine -\sqrt{2} and i\sqrt{2} to get \left(-1+i\right)\sqrt{2}.

$$-\sqrt{2}+i\sqrt{3}-\left(\left(1+i\right)\sqrt{3}+\left(-1+i\right)\sqrt{2}\right)+i\left(-\sqrt{2}\right)+\sqrt{3}$$

To find the opposite of \left(1+i\right)\sqrt{3}+\left(-1+i\right)\sqrt{2}, find the opposite of each term.

$$-\sqrt{2}+i\sqrt{3}+\left(-1-i\right)\sqrt{3}+\left(1-i\right)\sqrt{2}+i\left(-\sqrt{2}\right)+\sqrt{3}$$

Combine i\sqrt{3} and \left(-1-i\right)\sqrt{3} to get -\sqrt{3}.

$$-\sqrt{2}-\sqrt{3}+\left(1-i\right)\sqrt{2}+i\left(-\sqrt{2}\right)+\sqrt{3}$$

Combine -\sqrt{2} and i\left(-\sqrt{2}\right) to get \left(1+i\right)\left(-\sqrt{2}\right).

$$\left(1+i\right)\left(-\sqrt{2}\right)-\sqrt{3}+\left(1-i\right)\sqrt{2}+\sqrt{3}$$

Combine -\sqrt{3} and \sqrt{3} to get 0.

$$\left(1+i\right)\left(-\sqrt{2}\right)+\left(1-i\right)\sqrt{2}$$

Multiply 1+i and -1 to get -1-i.

$$\left(-1-i\right)\sqrt{2}+\left(1-i\right)\sqrt{2}$$

Combine \left(-1-i\right)\sqrt{2} and \left(1-i\right)\sqrt{2} to get -2i\sqrt{2}.

$$-2i\sqrt{2}$$

Real Part

$$0$$