Solve (sqrt{2}+i)(1+sqrt{2}i)

$$( \sqrt { 2 } + i ) ( 1 + \sqrt { 2 } i )$$

Evaluate

$$3i$$

Solution Steps

Apply the distributive property by multiplying each term of \sqrt{2}+i by each term of 1+i\sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Add 2i and i to get 3i.

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

Combine \sqrt{2} and -\sqrt{2} to get 0.

$$3i$$

Real Part

$$0$$

Solution Steps

Apply the distributive property by multiplying each term of \sqrt{2}+i by each term of 1+i\sqrt{2}.

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

The square of \sqrt{2} is 2.

$$Re(\sqrt{2}+2i+i-\sqrt{2})$$

Add 2i and i to get 3i.

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

Combine \sqrt{2} and -\sqrt{2} to get 0.

$$Re(3i)$$

The real part of 3i is 0.

$$0$$