Solve (1/5+1/8i)^-1

$$( \frac { 1 } { 5 } + \frac { 1 } { 8 i } ) ^ { - 1 }$$

Evaluate

$$\frac{320}{89}+\frac{200}{89}i\approx 3.595505618+2.247191011i$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{8i} by imaginary unit i.

$$\left(\frac{1}{5}+\frac{i}{-8}\right)^{-1}$$

Divide i by -8 to get -\frac{1}{8}i.

$$\left(\frac{1}{5}-\frac{1}{8}i\right)^{-1}$$

Calculate \frac{1}{5}-\frac{1}{8}i to the power of -1 and get \frac{320}{89}+\frac{200}{89}i.

$$\frac{320}{89}+\frac{200}{89}i$$

Real Part

$$\frac{320}{89} = 3\frac{53}{89} = 3.595505617977528$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{8i} by imaginary unit i.

$$Re(\left(\frac{1}{5}+\frac{i}{-8}\right)^{-1})$$

Divide i by -8 to get -\frac{1}{8}i.

$$Re(\left(\frac{1}{5}-\frac{1}{8}i\right)^{-1})$$

Calculate \frac{1}{5}-\frac{1}{8}i to the power of -1 and get \frac{320}{89}+\frac{200}{89}i.

$$Re(\frac{320}{89}+\frac{200}{89}i)$$

The real part of \frac{320}{89}+\frac{200}{89}i is \frac{320}{89}.

$$\frac{320}{89}$$