Solve 7-3i/4-3i

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

Evaluate

$$\frac{37}{25}+\frac{9}{25}i=1.48+0.36i$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$\frac{28+21i-12i+9}{25}$$

Combine the real and imaginary parts in 28+21i-12i+9.

$$\frac{28+9+\left(21-12\right)i}{25}$$

Do the additions in 28+9+\left(21-12\right)i.

$$\frac{37+9i}{25}$$

Divide 37+9i by 25 to get \frac{37}{25}+\frac{9}{25}i.

$$\frac{37}{25}+\frac{9}{25}i$$

Real Part

$$\frac{37}{25} = 1\frac{12}{25} = 1.48$$

Solution Steps

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

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

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

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

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

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

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

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

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

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

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

$$Re(\frac{28+21i-12i+9}{25})$$

Combine the real and imaginary parts in 28+21i-12i+9.

$$Re(\frac{28+9+\left(21-12\right)i}{25})$$

Do the additions in 28+9+\left(21-12\right)i.

$$Re(\frac{37+9i}{25})$$

Divide 37+9i by 25 to get \frac{37}{25}+\frac{9}{25}i.

$$Re(\frac{37}{25}+\frac{9}{25}i)$$

The real part of \frac{37}{25}+\frac{9}{25}i is \frac{37}{25}.

$$\frac{37}{25}$$