Solve 7-5i/4+3i

$$\frac { 7 - 5 i } { 4 + 3 i }$$

Evaluate

$$\frac{13}{25}-\frac{41}{25}i=0.52-1.64i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 4-3i.

$$\frac{\left(7-5i\right)\left(4-3i\right)}{\left(4+3i\right)\left(4-3i\right)}$$

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

$$\frac{\left(7-5i\right)\left(4-3i\right)}{4^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(7-5i\right)\left(4-3i\right)}{25}$$

Multiply complex numbers 7-5i and 4-3i like you multiply binomials.

$$\frac{7\times 4+7\times \left(-3i\right)-5i\times 4-5\left(-3\right)i^{2}}{25}$$

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

$$\frac{7\times 4+7\times \left(-3i\right)-5i\times 4-5\left(-3\right)\left(-1\right)}{25}$$

Do the multiplications in 7\times 4+7\times \left(-3i\right)-5i\times 4-5\left(-3\right)\left(-1\right).

$$\frac{28-21i-20i-15}{25}$$

Combine the real and imaginary parts in 28-21i-20i-15.

$$\frac{28-15+\left(-21-20\right)i}{25}$$

Do the additions in 28-15+\left(-21-20\right)i.

$$\frac{13-41i}{25}$$

Divide 13-41i by 25 to get \frac{13}{25}-\frac{41}{25}i.

$$\frac{13}{25}-\frac{41}{25}i$$

Real Part

$$\frac{13}{25} = 0.52$$

Solution Steps

Multiply both numerator and denominator of \frac{7-5i}{4+3i} by the complex conjugate of the denominator, 4-3i.

$$Re(\frac{\left(7-5i\right)\left(4-3i\right)}{\left(4+3i\right)\left(4-3i\right)})$$

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

$$Re(\frac{\left(7-5i\right)\left(4-3i\right)}{4^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(7-5i\right)\left(4-3i\right)}{25})$$

Multiply complex numbers 7-5i and 4-3i like you multiply binomials.

$$Re(\frac{7\times 4+7\times \left(-3i\right)-5i\times 4-5\left(-3\right)i^{2}}{25})$$

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

$$Re(\frac{7\times 4+7\times \left(-3i\right)-5i\times 4-5\left(-3\right)\left(-1\right)}{25})$$

Do the multiplications in 7\times 4+7\times \left(-3i\right)-5i\times 4-5\left(-3\right)\left(-1\right).

$$Re(\frac{28-21i-20i-15}{25})$$

Combine the real and imaginary parts in 28-21i-20i-15.

$$Re(\frac{28-15+\left(-21-20\right)i}{25})$$

Do the additions in 28-15+\left(-21-20\right)i.

$$Re(\frac{13-41i}{25})$$

Divide 13-41i by 25 to get \frac{13}{25}-\frac{41}{25}i.

$$Re(\frac{13}{25}-\frac{41}{25}i)$$

The real part of \frac{13}{25}-\frac{41}{25}i is \frac{13}{25}.

$$\frac{13}{25}$$