Solve 1/11-5i=

$$\frac { 1 } { 11 - 5 i } =$$

Evaluate

$$\frac{11}{146}+\frac{5}{146}i\approx 0.075342466+0.034246575i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 11+5i.

$$\frac{1\left(11+5i\right)}{\left(11-5i\right)\left(11+5i\right)}$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{1\left(11+5i\right)}{11^{2}-5^{2}i^{2}}$$

By definition, i^{2} is -1. Calculate the denominator.

$$\frac{1\left(11+5i\right)}{146}$$

Multiply 1 and 11+5i to get 11+5i.

$$\frac{11+5i}{146}$$

Divide 11+5i by 146 to get \frac{11}{146}+\frac{5}{146}i.

$$\frac{11}{146}+\frac{5}{146}i$$

Real Part

$$\frac{11}{146} = 0.07534246575342465$$

Solution Steps

Multiply both numerator and denominator of \frac{1}{11-5i} by the complex conjugate of the denominator, 11+5i.

$$Re(\frac{1\left(11+5i\right)}{\left(11-5i\right)\left(11+5i\right)})$$

Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$Re(\frac{1\left(11+5i\right)}{11^{2}-5^{2}i^{2}})$$

By definition, i^{2} is -1. Calculate the denominator.

$$Re(\frac{1\left(11+5i\right)}{146})$$

Multiply 1 and 11+5i to get 11+5i.

$$Re(\frac{11+5i}{146})$$

Divide 11+5i by 146 to get \frac{11}{146}+\frac{5}{146}i.

$$Re(\frac{11}{146}+\frac{5}{146}i)$$

The real part of \frac{11}{146}+\frac{5}{146}i is \frac{11}{146}.

$$\frac{11}{146}$$