Solve 10+10i/1-8i

$$\frac { 10 + 10 i } { 1 - 8 i }$$

Evaluate

$$-\frac{14}{13}+\frac{18}{13}i\approx -1.076923077+1.384615385i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 1+8i.

$$\frac{\left(10+10i\right)\left(1+8i\right)}{\left(1-8i\right)\left(1+8i\right)}$$

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

$$\frac{\left(10+10i\right)\left(1+8i\right)}{1^{2}-8^{2}i^{2}}$$

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

$$\frac{\left(10+10i\right)\left(1+8i\right)}{65}$$

Multiply complex numbers 10+10i and 1+8i like you multiply binomials.

$$\frac{10\times 1+10\times \left(8i\right)+10i\times 1+10\times 8i^{2}}{65}$$

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

$$\frac{10\times 1+10\times \left(8i\right)+10i\times 1+10\times 8\left(-1\right)}{65}$$

Do the multiplications in 10\times 1+10\times \left(8i\right)+10i\times 1+10\times 8\left(-1\right).

$$\frac{10+80i+10i-80}{65}$$

Combine the real and imaginary parts in 10+80i+10i-80.

$$\frac{10-80+\left(80+10\right)i}{65}$$

Do the additions in 10-80+\left(80+10\right)i.

$$\frac{-70+90i}{65}$$

Divide -70+90i by 65 to get -\frac{14}{13}+\frac{18}{13}i.

$$-\frac{14}{13}+\frac{18}{13}i$$

Real Part

$$-\frac{14}{13} = -1\frac{1}{13} = -1.0769230769230769$$

Solution Steps

Multiply both numerator and denominator of \frac{10+10i}{1-8i} by the complex conjugate of the denominator, 1+8i.

$$Re(\frac{\left(10+10i\right)\left(1+8i\right)}{\left(1-8i\right)\left(1+8i\right)})$$

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

$$Re(\frac{\left(10+10i\right)\left(1+8i\right)}{1^{2}-8^{2}i^{2}})$$

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

$$Re(\frac{\left(10+10i\right)\left(1+8i\right)}{65})$$

Multiply complex numbers 10+10i and 1+8i like you multiply binomials.

$$Re(\frac{10\times 1+10\times \left(8i\right)+10i\times 1+10\times 8i^{2}}{65})$$

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

$$Re(\frac{10\times 1+10\times \left(8i\right)+10i\times 1+10\times 8\left(-1\right)}{65})$$

Do the multiplications in 10\times 1+10\times \left(8i\right)+10i\times 1+10\times 8\left(-1\right).

$$Re(\frac{10+80i+10i-80}{65})$$

Combine the real and imaginary parts in 10+80i+10i-80.

$$Re(\frac{10-80+\left(80+10\right)i}{65})$$

Do the additions in 10-80+\left(80+10\right)i.

$$Re(\frac{-70+90i}{65})$$

Divide -70+90i by 65 to get -\frac{14}{13}+\frac{18}{13}i.

$$Re(-\frac{14}{13}+\frac{18}{13}i)$$

The real part of -\frac{14}{13}+\frac{18}{13}i is -\frac{14}{13}.

$$-\frac{14}{13}$$