Solve -1+5i/-8-7i

$$\frac { - 1 + 5 i } { - 8 - 7 i }$$

Evaluate

$$-\frac{27}{113}-\frac{47}{113}i\approx -0.238938053-0.415929204i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, -8+7i.

$$\frac{\left(-1+5i\right)\left(-8+7i\right)}{\left(-8-7i\right)\left(-8+7i\right)}$$

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

$$\frac{\left(-1+5i\right)\left(-8+7i\right)}{\left(-8\right)^{2}-7^{2}i^{2}}$$

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

$$\frac{\left(-1+5i\right)\left(-8+7i\right)}{113}$$

Multiply complex numbers -1+5i and -8+7i like you multiply binomials.

$$\frac{-\left(-8\right)-7i+5i\left(-8\right)+5\times 7i^{2}}{113}$$

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

$$\frac{-\left(-8\right)-7i+5i\left(-8\right)+5\times 7\left(-1\right)}{113}$$

Do the multiplications in -\left(-8\right)-7i+5i\left(-8\right)+5\times 7\left(-1\right).

$$\frac{8-7i-40i-35}{113}$$

Combine the real and imaginary parts in 8-7i-40i-35.

$$\frac{8-35+\left(-7-40\right)i}{113}$$

Do the additions in 8-35+\left(-7-40\right)i.

$$\frac{-27-47i}{113}$$

Divide -27-47i by 113 to get -\frac{27}{113}-\frac{47}{113}i.

$$-\frac{27}{113}-\frac{47}{113}i$$

Real Part

$$-\frac{27}{113} = -0.23893805309734514$$

Solution Steps

Multiply both numerator and denominator of \frac{-1+5i}{-8-7i} by the complex conjugate of the denominator, -8+7i.

$$Re(\frac{\left(-1+5i\right)\left(-8+7i\right)}{\left(-8-7i\right)\left(-8+7i\right)})$$

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

$$Re(\frac{\left(-1+5i\right)\left(-8+7i\right)}{\left(-8\right)^{2}-7^{2}i^{2}})$$

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

$$Re(\frac{\left(-1+5i\right)\left(-8+7i\right)}{113})$$

Multiply complex numbers -1+5i and -8+7i like you multiply binomials.

$$Re(\frac{-\left(-8\right)-7i+5i\left(-8\right)+5\times 7i^{2}}{113})$$

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

$$Re(\frac{-\left(-8\right)-7i+5i\left(-8\right)+5\times 7\left(-1\right)}{113})$$

Do the multiplications in -\left(-8\right)-7i+5i\left(-8\right)+5\times 7\left(-1\right).

$$Re(\frac{8-7i-40i-35}{113})$$

Combine the real and imaginary parts in 8-7i-40i-35.

$$Re(\frac{8-35+\left(-7-40\right)i}{113})$$

Do the additions in 8-35+\left(-7-40\right)i.

$$Re(\frac{-27-47i}{113})$$

Divide -27-47i by 113 to get -\frac{27}{113}-\frac{47}{113}i.

$$Re(-\frac{27}{113}-\frac{47}{113}i)$$

The real part of -\frac{27}{113}-\frac{47}{113}i is -\frac{27}{113}.

$$-\frac{27}{113}$$