Solve 7+i/-7+6i=

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

Evaluate

$$-\frac{43}{85}-\frac{49}{85}i\approx -0.505882353-0.576470588i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, -7-6i.

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

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

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

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

$$\frac{\left(7+i\right)\left(-7-6i\right)}{85}$$

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

$$\frac{7\left(-7\right)+7\times \left(-6i\right)-7i-6i^{2}}{85}$$

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

$$\frac{7\left(-7\right)+7\times \left(-6i\right)-7i-6\left(-1\right)}{85}$$

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

$$\frac{-49-42i-7i+6}{85}$$

Combine the real and imaginary parts in -49-42i-7i+6.

$$\frac{-49+6+\left(-42-7\right)i}{85}$$

Do the additions in -49+6+\left(-42-7\right)i.

$$\frac{-43-49i}{85}$$

Divide -43-49i by 85 to get -\frac{43}{85}-\frac{49}{85}i.

$$-\frac{43}{85}-\frac{49}{85}i$$

Real Part

$$-\frac{43}{85} = -0.5058823529411764$$

Solution Steps

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

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

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

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

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

$$Re(\frac{\left(7+i\right)\left(-7-6i\right)}{85})$$

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

$$Re(\frac{7\left(-7\right)+7\times \left(-6i\right)-7i-6i^{2}}{85})$$

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

$$Re(\frac{7\left(-7\right)+7\times \left(-6i\right)-7i-6\left(-1\right)}{85})$$

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

$$Re(\frac{-49-42i-7i+6}{85})$$

Combine the real and imaginary parts in -49-42i-7i+6.

$$Re(\frac{-49+6+\left(-42-7\right)i}{85})$$

Do the additions in -49+6+\left(-42-7\right)i.

$$Re(\frac{-43-49i}{85})$$

Divide -43-49i by 85 to get -\frac{43}{85}-\frac{49}{85}i.

$$Re(-\frac{43}{85}-\frac{49}{85}i)$$

The real part of -\frac{43}{85}-\frac{49}{85}i is -\frac{43}{85}.

$$-\frac{43}{85}$$