Solve i(1+4i)^3/(3-2i)^2

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

Evaluate

$$\frac{824}{169}+\frac{389}{169}i\approx 4.875739645+2.301775148i$$

Solution Steps

Calculate 1+4i to the power of 3 and get -47-52i.

$$\frac{i\left(-47-52i\right)}{\left(3-2i\right)^{2}}$$

Multiply i and -47-52i to get 52-47i.

$$\frac{52-47i}{\left(3-2i\right)^{2}}$$

Calculate 3-2i to the power of 2 and get 5-12i.

$$\frac{52-47i}{5-12i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 5+12i.

$$\frac{\left(52-47i\right)\left(5+12i\right)}{\left(5-12i\right)\left(5+12i\right)}$$

Do the multiplications in \frac{\left(52-47i\right)\left(5+12i\right)}{\left(5-12i\right)\left(5+12i\right)}.

$$\frac{824+389i}{169}$$

Divide 824+389i by 169 to get \frac{824}{169}+\frac{389}{169}i.

$$\frac{824}{169}+\frac{389}{169}i$$

Real Part

$$\frac{824}{169} = 4\frac{148}{169} = 4.875739644970414$$

Solution Steps

Calculate 1+4i to the power of 3 and get -47-52i.

$$Re(\frac{i\left(-47-52i\right)}{\left(3-2i\right)^{2}})$$

Multiply i and -47-52i to get 52-47i.

$$Re(\frac{52-47i}{\left(3-2i\right)^{2}})$$

Calculate 3-2i to the power of 2 and get 5-12i.

$$Re(\frac{52-47i}{5-12i})$$

Multiply both numerator and denominator of \frac{52-47i}{5-12i} by the complex conjugate of the denominator, 5+12i.

$$Re(\frac{\left(52-47i\right)\left(5+12i\right)}{\left(5-12i\right)\left(5+12i\right)})$$

Do the multiplications in \frac{\left(52-47i\right)\left(5+12i\right)}{\left(5-12i\right)\left(5+12i\right)}.

$$Re(\frac{824+389i}{169})$$

Divide 824+389i by 169 to get \frac{824}{169}+\frac{389}{169}i.

$$Re(\frac{824}{169}+\frac{389}{169}i)$$

The real part of \frac{824}{169}+\frac{389}{169}i is \frac{824}{169}.

$$\frac{824}{169}$$