Solve -5+i/-6+6i

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

Evaluate

$$\frac{1}{2}+\frac{1}{3}i\approx 0.5+0.333333333i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, -6-6i.

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

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

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

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

$$\frac{\left(-5+i\right)\left(-6-6i\right)}{72}$$

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

$$\frac{-5\left(-6\right)-5\times \left(-6i\right)-6i-6i^{2}}{72}$$

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

$$\frac{-5\left(-6\right)-5\times \left(-6i\right)-6i-6\left(-1\right)}{72}$$

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

$$\frac{30+30i-6i+6}{72}$$

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

$$\frac{30+6+\left(30-6\right)i}{72}$$

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

$$\frac{36+24i}{72}$$

Divide 36+24i by 72 to get \frac{1}{2}+\frac{1}{3}i.

$$\frac{1}{2}+\frac{1}{3}i$$

Real Part

$$\frac{1}{2} = 0.5$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(-5+i\right)\left(-6-6i\right)}{72})$$

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

$$Re(\frac{-5\left(-6\right)-5\times \left(-6i\right)-6i-6i^{2}}{72})$$

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

$$Re(\frac{-5\left(-6\right)-5\times \left(-6i\right)-6i-6\left(-1\right)}{72})$$

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

$$Re(\frac{30+30i-6i+6}{72})$$

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

$$Re(\frac{30+6+\left(30-6\right)i}{72})$$

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

$$Re(\frac{36+24i}{72})$$

Divide 36+24i by 72 to get \frac{1}{2}+\frac{1}{3}i.

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

The real part of \frac{1}{2}+\frac{1}{3}i is \frac{1}{2}.

$$\frac{1}{2}$$