Solve 9-5i/5+7i

$$\frac { 9 - 5 i } { 5 + 7 i }$$

Evaluate

$$\frac{5}{37}-\frac{44}{37}i\approx 0.135135135-1.189189189i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 5-7i.

$$\frac{\left(9-5i\right)\left(5-7i\right)}{\left(5+7i\right)\left(5-7i\right)}$$

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

$$\frac{\left(9-5i\right)\left(5-7i\right)}{5^{2}-7^{2}i^{2}}$$

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

$$\frac{\left(9-5i\right)\left(5-7i\right)}{74}$$

Multiply complex numbers 9-5i and 5-7i like you multiply binomials.

$$\frac{9\times 5+9\times \left(-7i\right)-5i\times 5-5\left(-7\right)i^{2}}{74}$$

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

$$\frac{9\times 5+9\times \left(-7i\right)-5i\times 5-5\left(-7\right)\left(-1\right)}{74}$$

Do the multiplications in 9\times 5+9\times \left(-7i\right)-5i\times 5-5\left(-7\right)\left(-1\right).

$$\frac{45-63i-25i-35}{74}$$

Combine the real and imaginary parts in 45-63i-25i-35.

$$\frac{45-35+\left(-63-25\right)i}{74}$$

Do the additions in 45-35+\left(-63-25\right)i.

$$\frac{10-88i}{74}$$

Divide 10-88i by 74 to get \frac{5}{37}-\frac{44}{37}i.

$$\frac{5}{37}-\frac{44}{37}i$$

Real Part

$$\frac{5}{37} = 0.13513513513513514$$

Solution Steps

Multiply both numerator and denominator of \frac{9-5i}{5+7i} by the complex conjugate of the denominator, 5-7i.

$$Re(\frac{\left(9-5i\right)\left(5-7i\right)}{\left(5+7i\right)\left(5-7i\right)})$$

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

$$Re(\frac{\left(9-5i\right)\left(5-7i\right)}{5^{2}-7^{2}i^{2}})$$

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

$$Re(\frac{\left(9-5i\right)\left(5-7i\right)}{74})$$

Multiply complex numbers 9-5i and 5-7i like you multiply binomials.

$$Re(\frac{9\times 5+9\times \left(-7i\right)-5i\times 5-5\left(-7\right)i^{2}}{74})$$

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

$$Re(\frac{9\times 5+9\times \left(-7i\right)-5i\times 5-5\left(-7\right)\left(-1\right)}{74})$$

Do the multiplications in 9\times 5+9\times \left(-7i\right)-5i\times 5-5\left(-7\right)\left(-1\right).

$$Re(\frac{45-63i-25i-35}{74})$$

Combine the real and imaginary parts in 45-63i-25i-35.

$$Re(\frac{45-35+\left(-63-25\right)i}{74})$$

Do the additions in 45-35+\left(-63-25\right)i.

$$Re(\frac{10-88i}{74})$$

Divide 10-88i by 74 to get \frac{5}{37}-\frac{44}{37}i.

$$Re(\frac{5}{37}-\frac{44}{37}i)$$

The real part of \frac{5}{37}-\frac{44}{37}i is \frac{5}{37}.

$$\frac{5}{37}$$