Solve |3-2i/3+2i|:|(2+3i)^2|

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

Evaluate

$$\frac{1}{13}\approx 0.076923077$$

Solution Steps

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

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

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

$$\frac{|\frac{5-12i}{13}|}{\left(|2+3i|\right)^{2}}$$

Divide 5-12i by 13 to get \frac{5}{13}-\frac{12}{13}i.

$$\frac{|\frac{5}{13}-\frac{12}{13}i|}{\left(|2+3i|\right)^{2}}$$

The modulus of a complex number a+bi is \sqrt{a^{2}+b^{2}}. The modulus of \frac{5}{13}-\frac{12}{13}i is 1.

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

The modulus of a complex number a+bi is \sqrt{a^{2}+b^{2}}. The modulus of 2+3i is \sqrt{13}.

$$\frac{1}{\left(\sqrt{13}\right)^{2}}$$

The square of \sqrt{13} is 13.

$$\frac{1}{13}$$

Real Part

$$\frac{1}{13} = 0.07692307692307693$$

Solution Steps

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

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

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

$$Re(\frac{|\frac{5-12i}{13}|}{\left(|2+3i|\right)^{2}})$$

Divide 5-12i by 13 to get \frac{5}{13}-\frac{12}{13}i.

$$Re(\frac{|\frac{5}{13}-\frac{12}{13}i|}{\left(|2+3i|\right)^{2}})$$

The modulus of a complex number a+bi is \sqrt{a^{2}+b^{2}}. The modulus of \frac{5}{13}-\frac{12}{13}i is 1.

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

The modulus of a complex number a+bi is \sqrt{a^{2}+b^{2}}. The modulus of 2+3i is \sqrt{13}.

$$Re(\frac{1}{\left(\sqrt{13}\right)^{2}})$$

The square of \sqrt{13} is 13.

$$Re(\frac{1}{13})$$

The real part of \frac{1}{13} is \frac{1}{13}.

$$\frac{1}{13}$$