Solve -10-5i/-6+6i

$$\frac { - 10 - 5 i } { - 6 + 6 i }$$

Evaluate

$$\frac{5}{12}+\frac{5}{4}i\approx 0.416666667+1.25i$$

Solution Steps

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

$$\frac{\left(-10-5i\right)\left(-6-6i\right)}{\left(-6+6i\right)\left(-6-6i\right)}$$

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

$$\frac{\left(-10-5i\right)\left(-6-6i\right)}{\left(-6\right)^{2}-6^{2}i^{2}}$$

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

$$\frac{\left(-10-5i\right)\left(-6-6i\right)}{72}$$

Multiply complex numbers -10-5i and -6-6i like you multiply binomials.

$$\frac{-10\left(-6\right)-10\times \left(-6i\right)-5i\left(-6\right)-5\left(-6\right)i^{2}}{72}$$

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

$$\frac{-10\left(-6\right)-10\times \left(-6i\right)-5i\left(-6\right)-5\left(-6\right)\left(-1\right)}{72}$$

Do the multiplications in -10\left(-6\right)-10\times \left(-6i\right)-5i\left(-6\right)-5\left(-6\right)\left(-1\right).

$$\frac{60+60i+30i-30}{72}$$

Combine the real and imaginary parts in 60+60i+30i-30.

$$\frac{60-30+\left(60+30\right)i}{72}$$

Do the additions in 60-30+\left(60+30\right)i.

$$\frac{30+90i}{72}$$

Divide 30+90i by 72 to get \frac{5}{12}+\frac{5}{4}i.

$$\frac{5}{12}+\frac{5}{4}i$$

Real Part

$$\frac{5}{12} = 0.4166666666666667$$

Solution Steps

Multiply both numerator and denominator of \frac{-10-5i}{-6+6i} by the complex conjugate of the denominator, -6-6i.

$$Re(\frac{\left(-10-5i\right)\left(-6-6i\right)}{\left(-6+6i\right)\left(-6-6i\right)})$$

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

$$Re(\frac{\left(-10-5i\right)\left(-6-6i\right)}{\left(-6\right)^{2}-6^{2}i^{2}})$$

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

$$Re(\frac{\left(-10-5i\right)\left(-6-6i\right)}{72})$$

Multiply complex numbers -10-5i and -6-6i like you multiply binomials.

$$Re(\frac{-10\left(-6\right)-10\times \left(-6i\right)-5i\left(-6\right)-5\left(-6\right)i^{2}}{72})$$

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

$$Re(\frac{-10\left(-6\right)-10\times \left(-6i\right)-5i\left(-6\right)-5\left(-6\right)\left(-1\right)}{72})$$

Do the multiplications in -10\left(-6\right)-10\times \left(-6i\right)-5i\left(-6\right)-5\left(-6\right)\left(-1\right).

$$Re(\frac{60+60i+30i-30}{72})$$

Combine the real and imaginary parts in 60+60i+30i-30.

$$Re(\frac{60-30+\left(60+30\right)i}{72})$$

Do the additions in 60-30+\left(60+30\right)i.

$$Re(\frac{30+90i}{72})$$

Divide 30+90i by 72 to get \frac{5}{12}+\frac{5}{4}i.

$$Re(\frac{5}{12}+\frac{5}{4}i)$$

The real part of \frac{5}{12}+\frac{5}{4}i is \frac{5}{12}.

$$\frac{5}{12}$$