Solve i^97/3+2i

$$\frac { i ^ { 97 } } { 3 + 2 i }$$

Evaluate

$$\frac{2}{13}+\frac{3}{13}i\approx 0.153846154+0.230769231i$$

Solution Steps

Calculate i to the power of 97 and get i.

$$\frac{i}{3+2i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 3-2i.

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

Do the multiplications in \frac{i\left(3-2i\right)}{\left(3+2i\right)\left(3-2i\right)}.

$$\frac{2+3i}{13}$$

Divide 2+3i by 13 to get \frac{2}{13}+\frac{3}{13}i.

$$\frac{2}{13}+\frac{3}{13}i$$

Real Part

$$\frac{2}{13} = 0.15384615384615385$$

Solution Steps

Calculate i to the power of 97 and get i.

$$Re(\frac{i}{3+2i})$$

Multiply both numerator and denominator of \frac{i}{3+2i} by the complex conjugate of the denominator, 3-2i.

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

Do the multiplications in \frac{i\left(3-2i\right)}{\left(3+2i\right)\left(3-2i\right)}.

$$Re(\frac{2+3i}{13})$$

Divide 2+3i by 13 to get \frac{2}{13}+\frac{3}{13}i.

$$Re(\frac{2}{13}+\frac{3}{13}i)$$

The real part of \frac{2}{13}+\frac{3}{13}i is \frac{2}{13}.

$$\frac{2}{13}$$