Solve left(1-2iright)^2+1-2i+1/left(1-2iright)^3-1

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

Evaluate

$$\frac{1}{2}i=0.5i$$

Solution Steps

Do the additions in 1-2i+1.

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

Calculate 1-2i to the power of 2 and get -3-4i.

$$\frac{-3-4i+2-2i}{\left(1-2i\right)^{3}-1}$$

Do the additions in -3-4i+2-2i.

$$\frac{-1-6i}{\left(1-2i\right)^{3}-1}$$

Calculate 1-2i to the power of 3 and get -11+2i.

$$\frac{-1-6i}{-11+2i-1}$$

Subtract 1 from -11+2i to get -12+2i.

$$\frac{-1-6i}{-12+2i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, -12-2i.

$$\frac{\left(-1-6i\right)\left(-12-2i\right)}{\left(-12+2i\right)\left(-12-2i\right)}$$

Do the multiplications in \frac{\left(-1-6i\right)\left(-12-2i\right)}{\left(-12+2i\right)\left(-12-2i\right)}.

$$\frac{74i}{148}$$

Divide 74i by 148 to get \frac{1}{2}i.

$$\frac{1}{2}i$$

Real Part

$$0$$

Solution Steps

Do the additions in 1-2i+1.

$$Re(\frac{\left(1-2i\right)^{2}+2-2i}{\left(1-2i\right)^{3}-1})$$

Calculate 1-2i to the power of 2 and get -3-4i.

$$Re(\frac{-3-4i+2-2i}{\left(1-2i\right)^{3}-1})$$

Do the additions in -3-4i+2-2i.

$$Re(\frac{-1-6i}{\left(1-2i\right)^{3}-1})$$

Calculate 1-2i to the power of 3 and get -11+2i.

$$Re(\frac{-1-6i}{-11+2i-1})$$

Subtract 1 from -11+2i to get -12+2i.

$$Re(\frac{-1-6i}{-12+2i})$$

Multiply both numerator and denominator of \frac{-1-6i}{-12+2i} by the complex conjugate of the denominator, -12-2i.

$$Re(\frac{\left(-1-6i\right)\left(-12-2i\right)}{\left(-12+2i\right)\left(-12-2i\right)})$$

Do the multiplications in \frac{\left(-1-6i\right)\left(-12-2i\right)}{\left(-12+2i\right)\left(-12-2i\right)}.

$$Re(\frac{74i}{148})$$

Divide 74i by 148 to get \frac{1}{2}i.

$$Re(\frac{1}{2}i)$$

The real part of \frac{1}{2}i is 0.

$$0$$