Solve 6+7i/-5-6i

$$\frac { 6 + 7 i } { - 5 - 6 i }$$

Evaluate

$$-\frac{72}{61}+\frac{1}{61}i\approx -1.180327869+0.016393443i$$

Solution Steps

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

$$\frac{\left(6+7i\right)\left(-5+6i\right)}{\left(-5-6i\right)\left(-5+6i\right)}$$

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

$$\frac{\left(6+7i\right)\left(-5+6i\right)}{\left(-5\right)^{2}-6^{2}i^{2}}$$

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

$$\frac{\left(6+7i\right)\left(-5+6i\right)}{61}$$

Multiply complex numbers 6+7i and -5+6i like you multiply binomials.

$$\frac{6\left(-5\right)+6\times \left(6i\right)+7i\left(-5\right)+7\times 6i^{2}}{61}$$

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

$$\frac{6\left(-5\right)+6\times \left(6i\right)+7i\left(-5\right)+7\times 6\left(-1\right)}{61}$$

Do the multiplications in 6\left(-5\right)+6\times \left(6i\right)+7i\left(-5\right)+7\times 6\left(-1\right).

$$\frac{-30+36i-35i-42}{61}$$

Combine the real and imaginary parts in -30+36i-35i-42.

$$\frac{-30-42+\left(36-35\right)i}{61}$$

Do the additions in -30-42+\left(36-35\right)i.

$$\frac{-72+i}{61}$$

Divide -72+i by 61 to get -\frac{72}{61}+\frac{1}{61}i.

$$-\frac{72}{61}+\frac{1}{61}i$$

Real Part

$$-\frac{72}{61} = -1\frac{11}{61} = -1.180327868852459$$

Solution Steps

Multiply both numerator and denominator of \frac{6+7i}{-5-6i} by the complex conjugate of the denominator, -5+6i.

$$Re(\frac{\left(6+7i\right)\left(-5+6i\right)}{\left(-5-6i\right)\left(-5+6i\right)})$$

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

$$Re(\frac{\left(6+7i\right)\left(-5+6i\right)}{\left(-5\right)^{2}-6^{2}i^{2}})$$

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

$$Re(\frac{\left(6+7i\right)\left(-5+6i\right)}{61})$$

Multiply complex numbers 6+7i and -5+6i like you multiply binomials.

$$Re(\frac{6\left(-5\right)+6\times \left(6i\right)+7i\left(-5\right)+7\times 6i^{2}}{61})$$

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

$$Re(\frac{6\left(-5\right)+6\times \left(6i\right)+7i\left(-5\right)+7\times 6\left(-1\right)}{61})$$

Do the multiplications in 6\left(-5\right)+6\times \left(6i\right)+7i\left(-5\right)+7\times 6\left(-1\right).

$$Re(\frac{-30+36i-35i-42}{61})$$

Combine the real and imaginary parts in -30+36i-35i-42.

$$Re(\frac{-30-42+\left(36-35\right)i}{61})$$

Do the additions in -30-42+\left(36-35\right)i.

$$Re(\frac{-72+i}{61})$$

Divide -72+i by 61 to get -\frac{72}{61}+\frac{1}{61}i.

$$Re(-\frac{72}{61}+\frac{1}{61}i)$$

The real part of -\frac{72}{61}+\frac{1}{61}i is -\frac{72}{61}.

$$-\frac{72}{61}$$