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

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

Evaluate

$$-\frac{3}{37}+\frac{18}{37}i\approx -0.081081081+0.486486486i$$

Solution Steps

Do the additions in 2-2i+1.

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

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

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

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

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

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

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

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

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

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

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

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

$$\frac{-12+72i}{148}$$

Divide -12+72i by 148 to get -\frac{3}{37}+\frac{18}{37}i.

$$-\frac{3}{37}+\frac{18}{37}i$$

Real Part

$$-\frac{3}{37} = -0.08108108108108109$$

Solution Steps

Do the additions in 2-2i+1.

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

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{-12+72i}{148})$$

Divide -12+72i by 148 to get -\frac{3}{37}+\frac{18}{37}i.

$$Re(-\frac{3}{37}+\frac{18}{37}i)$$

The real part of -\frac{3}{37}+\frac{18}{37}i is -\frac{3}{37}.

$$-\frac{3}{37}$$