Solve 8-7i/8+7i

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

Evaluate

$$\frac{15}{113}-\frac{112}{113}i\approx 0.132743363-0.991150442i$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{64-56i-56i-49}{113}$$

Combine the real and imaginary parts in 64-56i-56i-49.

$$\frac{64-49+\left(-56-56\right)i}{113}$$

Do the additions in 64-49+\left(-56-56\right)i.

$$\frac{15-112i}{113}$$

Divide 15-112i by 113 to get \frac{15}{113}-\frac{112}{113}i.

$$\frac{15}{113}-\frac{112}{113}i$$

Real Part

$$\frac{15}{113} = 0.13274336283185842$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{64-56i-56i-49}{113})$$

Combine the real and imaginary parts in 64-56i-56i-49.

$$Re(\frac{64-49+\left(-56-56\right)i}{113})$$

Do the additions in 64-49+\left(-56-56\right)i.

$$Re(\frac{15-112i}{113})$$

Divide 15-112i by 113 to get \frac{15}{113}-\frac{112}{113}i.

$$Re(\frac{15}{113}-\frac{112}{113}i)$$

The real part of \frac{15}{113}-\frac{112}{113}i is \frac{15}{113}.

$$\frac{15}{113}$$