Solve 5-3i/6+5i

$$\frac { 5 - 3 i } { 6 + 5 i }$$

Evaluate

$$\frac{15}{61}-\frac{43}{61}i\approx 0.245901639-0.704918033i$$

Solution Steps

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

$$\frac{\left(5-3i\right)\left(6-5i\right)}{\left(6+5i\right)\left(6-5i\right)}$$

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

$$\frac{\left(5-3i\right)\left(6-5i\right)}{6^{2}-5^{2}i^{2}}$$

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

$$\frac{\left(5-3i\right)\left(6-5i\right)}{61}$$

Multiply complex numbers 5-3i and 6-5i like you multiply binomials.

$$\frac{5\times 6+5\times \left(-5i\right)-3i\times 6-3\left(-5\right)i^{2}}{61}$$

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

$$\frac{5\times 6+5\times \left(-5i\right)-3i\times 6-3\left(-5\right)\left(-1\right)}{61}$$

Do the multiplications in 5\times 6+5\times \left(-5i\right)-3i\times 6-3\left(-5\right)\left(-1\right).

$$\frac{30-25i-18i-15}{61}$$

Combine the real and imaginary parts in 30-25i-18i-15.

$$\frac{30-15+\left(-25-18\right)i}{61}$$

Do the additions in 30-15+\left(-25-18\right)i.

$$\frac{15-43i}{61}$$

Divide 15-43i by 61 to get \frac{15}{61}-\frac{43}{61}i.

$$\frac{15}{61}-\frac{43}{61}i$$

Real Part

$$\frac{15}{61} = 0.2459016393442623$$

Solution Steps

Multiply both numerator and denominator of \frac{5-3i}{6+5i} by the complex conjugate of the denominator, 6-5i.

$$Re(\frac{\left(5-3i\right)\left(6-5i\right)}{\left(6+5i\right)\left(6-5i\right)})$$

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

$$Re(\frac{\left(5-3i\right)\left(6-5i\right)}{6^{2}-5^{2}i^{2}})$$

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

$$Re(\frac{\left(5-3i\right)\left(6-5i\right)}{61})$$

Multiply complex numbers 5-3i and 6-5i like you multiply binomials.

$$Re(\frac{5\times 6+5\times \left(-5i\right)-3i\times 6-3\left(-5\right)i^{2}}{61})$$

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

$$Re(\frac{5\times 6+5\times \left(-5i\right)-3i\times 6-3\left(-5\right)\left(-1\right)}{61})$$

Do the multiplications in 5\times 6+5\times \left(-5i\right)-3i\times 6-3\left(-5\right)\left(-1\right).

$$Re(\frac{30-25i-18i-15}{61})$$

Combine the real and imaginary parts in 30-25i-18i-15.

$$Re(\frac{30-15+\left(-25-18\right)i}{61})$$

Do the additions in 30-15+\left(-25-18\right)i.

$$Re(\frac{15-43i}{61})$$

Divide 15-43i by 61 to get \frac{15}{61}-\frac{43}{61}i.

$$Re(\frac{15}{61}-\frac{43}{61}i)$$

The real part of \frac{15}{61}-\frac{43}{61}i is \frac{15}{61}.

$$\frac{15}{61}$$