Solve -6-i/1+5i

$$\frac { - 6 - i } { 1 + 5 i }$$

Evaluate

$$-\frac{11}{26}+\frac{29}{26}i\approx -0.423076923+1.115384615i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1-5i.

$$\frac{\left(-6-i\right)\left(1-5i\right)}{\left(1+5i\right)\left(1-5i\right)}$$

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

$$\frac{\left(-6-i\right)\left(1-5i\right)}{1^{2}-5^{2}i^{2}}$$

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

$$\frac{\left(-6-i\right)\left(1-5i\right)}{26}$$

Multiply complex numbers -6-i and 1-5i like you multiply binomials.

$$\frac{-6-6\times \left(-5i\right)-i-\left(-5i^{2}\right)}{26}$$

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

$$\frac{-6-6\times \left(-5i\right)-i-\left(-5\left(-1\right)\right)}{26}$$

Do the multiplications in -6-6\times \left(-5i\right)-i-\left(-5\left(-1\right)\right).

$$\frac{-6+30i-i-5}{26}$$

Combine the real and imaginary parts in -6+30i-i-5.

$$\frac{-6-5+\left(30-1\right)i}{26}$$

Do the additions in -6-5+\left(30-1\right)i.

$$\frac{-11+29i}{26}$$

Divide -11+29i by 26 to get -\frac{11}{26}+\frac{29}{26}i.

$$-\frac{11}{26}+\frac{29}{26}i$$

Real Part

$$-\frac{11}{26} = -0.4230769230769231$$

Solution Steps

Multiply both numerator and denominator of \frac{-6-i}{1+5i} by the complex conjugate of the denominator, 1-5i.

$$Re(\frac{\left(-6-i\right)\left(1-5i\right)}{\left(1+5i\right)\left(1-5i\right)})$$

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

$$Re(\frac{\left(-6-i\right)\left(1-5i\right)}{1^{2}-5^{2}i^{2}})$$

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

$$Re(\frac{\left(-6-i\right)\left(1-5i\right)}{26})$$

Multiply complex numbers -6-i and 1-5i like you multiply binomials.

$$Re(\frac{-6-6\times \left(-5i\right)-i-\left(-5i^{2}\right)}{26})$$

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

$$Re(\frac{-6-6\times \left(-5i\right)-i-\left(-5\left(-1\right)\right)}{26})$$

Do the multiplications in -6-6\times \left(-5i\right)-i-\left(-5\left(-1\right)\right).

$$Re(\frac{-6+30i-i-5}{26})$$

Combine the real and imaginary parts in -6+30i-i-5.

$$Re(\frac{-6-5+\left(30-1\right)i}{26})$$

Do the additions in -6-5+\left(30-1\right)i.

$$Re(\frac{-11+29i}{26})$$

Divide -11+29i by 26 to get -\frac{11}{26}+\frac{29}{26}i.

$$Re(-\frac{11}{26}+\frac{29}{26}i)$$

The real part of -\frac{11}{26}+\frac{29}{26}i is -\frac{11}{26}.

$$-\frac{11}{26}$$