Solve (1+1)^2/1+2i+(1-1)^2/1-2i

$$\frac { ( 1 + 1 ) ^ { 2 } } { 1 + 2 i } + \frac { ( 1 - 1 ) ^ { 2 } } { 1 - 2 i }$$

Evaluate

$$\frac{4}{5}-\frac{8}{5}i=0.8-1.6i$$

Solution Steps

Add 1 and 1 to get 2.

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

Calculate 2 to the power of 2 and get 4.

$$\frac{4}{1+2i}+\frac{\left(1-1\right)^{2}}{1-2i}$$

Multiply both numerator and denominator of \frac{4}{1+2i} by the complex conjugate of the denominator, 1-2i.

$$\frac{4\left(1-2i\right)}{\left(1+2i\right)\left(1-2i\right)}+\frac{\left(1-1\right)^{2}}{1-2i}$$

Do the multiplications in \frac{4\left(1-2i\right)}{\left(1+2i\right)\left(1-2i\right)}.

$$\frac{4-8i}{5}+\frac{\left(1-1\right)^{2}}{1-2i}$$

Divide 4-8i by 5 to get \frac{4}{5}-\frac{8}{5}i.

$$\frac{4}{5}-\frac{8}{5}i+\frac{\left(1-1\right)^{2}}{1-2i}$$

Subtract 1 from 1 to get 0.

$$\frac{4}{5}-\frac{8}{5}i+\frac{0^{2}}{1-2i}$$

Calculate 0 to the power of 2 and get 0.

$$\frac{4}{5}-\frac{8}{5}i+\frac{0}{1-2i}$$

Zero divided by any non-zero number gives zero.

$$\frac{4}{5}-\frac{8}{5}i+0$$

Anything plus zero gives itself.

$$\frac{4}{5}-\frac{8}{5}i$$

Real Part

$$\frac{4}{5} = 0.8$$

Solution Steps

Add 1 and 1 to get 2.

$$Re(\frac{2^{2}}{1+2i}+\frac{\left(1-1\right)^{2}}{1-2i})$$

Calculate 2 to the power of 2 and get 4.

$$Re(\frac{4}{1+2i}+\frac{\left(1-1\right)^{2}}{1-2i})$$

Multiply both numerator and denominator of \frac{4}{1+2i} by the complex conjugate of the denominator, 1-2i.

$$Re(\frac{4\left(1-2i\right)}{\left(1+2i\right)\left(1-2i\right)}+\frac{\left(1-1\right)^{2}}{1-2i})$$

Do the multiplications in \frac{4\left(1-2i\right)}{\left(1+2i\right)\left(1-2i\right)}.

$$Re(\frac{4-8i}{5}+\frac{\left(1-1\right)^{2}}{1-2i})$$

Divide 4-8i by 5 to get \frac{4}{5}-\frac{8}{5}i.

$$Re(\frac{4}{5}-\frac{8}{5}i+\frac{\left(1-1\right)^{2}}{1-2i})$$

Subtract 1 from 1 to get 0.

$$Re(\frac{4}{5}-\frac{8}{5}i+\frac{0^{2}}{1-2i})$$

Calculate 0 to the power of 2 and get 0.

$$Re(\frac{4}{5}-\frac{8}{5}i+\frac{0}{1-2i})$$

Zero divided by any non-zero number gives zero.

$$Re(\frac{4}{5}-\frac{8}{5}i+0)$$

Anything plus zero gives itself.

$$Re(\frac{4}{5}-\frac{8}{5}i)$$

The real part of \frac{4}{5}-\frac{8}{5}i is \frac{4}{5}.

$$\frac{4}{5}$$