Solve 7-6i/7+6i

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

Evaluate

$$\frac{13}{85}-\frac{84}{85}i\approx 0.152941176-0.988235294i$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{49-42i-42i-36}{85}$$

Combine the real and imaginary parts in 49-42i-42i-36.

$$\frac{49-36+\left(-42-42\right)i}{85}$$

Do the additions in 49-36+\left(-42-42\right)i.

$$\frac{13-84i}{85}$$

Divide 13-84i by 85 to get \frac{13}{85}-\frac{84}{85}i.

$$\frac{13}{85}-\frac{84}{85}i$$

Real Part

$$\frac{13}{85} = 0.15294117647058825$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{49-42i-42i-36}{85})$$

Combine the real and imaginary parts in 49-42i-42i-36.

$$Re(\frac{49-36+\left(-42-42\right)i}{85})$$

Do the additions in 49-36+\left(-42-42\right)i.

$$Re(\frac{13-84i}{85})$$

Divide 13-84i by 85 to get \frac{13}{85}-\frac{84}{85}i.

$$Re(\frac{13}{85}-\frac{84}{85}i)$$

The real part of \frac{13}{85}-\frac{84}{85}i is \frac{13}{85}.

$$\frac{13}{85}$$