Solve (1/5+3i)^2

$$( \frac { 1 } { 5 + 3 i } ) ^ { 2 }$$

Evaluate

$$\frac{4}{289}-\frac{15}{578}i\approx 0.01384083-0.025951557i$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{5+3i} by the complex conjugate of the denominator, 5-3i.

$$\left(\frac{1\left(5-3i\right)}{\left(5+3i\right)\left(5-3i\right)}\right)^{2}$$

Do the multiplications in \frac{1\left(5-3i\right)}{\left(5+3i\right)\left(5-3i\right)}.

$$\left(\frac{5-3i}{34}\right)^{2}$$

Divide 5-3i by 34 to get \frac{5}{34}-\frac{3}{34}i.

$$\left(\frac{5}{34}-\frac{3}{34}i\right)^{2}$$

Calculate \frac{5}{34}-\frac{3}{34}i to the power of 2 and get \frac{4}{289}-\frac{15}{578}i.

$$\frac{4}{289}-\frac{15}{578}i$$

Real Part

$$\frac{4}{289} = 0.01384083044982699$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{5+3i} by the complex conjugate of the denominator, 5-3i.

$$Re(\left(\frac{1\left(5-3i\right)}{\left(5+3i\right)\left(5-3i\right)}\right)^{2})$$

Do the multiplications in \frac{1\left(5-3i\right)}{\left(5+3i\right)\left(5-3i\right)}.

$$Re(\left(\frac{5-3i}{34}\right)^{2})$$

Divide 5-3i by 34 to get \frac{5}{34}-\frac{3}{34}i.

$$Re(\left(\frac{5}{34}-\frac{3}{34}i\right)^{2})$$

Calculate \frac{5}{34}-\frac{3}{34}i to the power of 2 and get \frac{4}{289}-\frac{15}{578}i.

$$Re(\frac{4}{289}-\frac{15}{578}i)$$

The real part of \frac{4}{289}-\frac{15}{578}i is \frac{4}{289}.

$$\frac{4}{289}$$