Solve 2+5i/7-3i

$$\frac { 2 + 5 i } { 7 - 3 i }$$

Evaluate

$$-\frac{1}{58}+\frac{41}{58}i\approx -0.017241379+0.706896552i$$

Solution Steps

Multiply both numerator and denominator by the complex conjugate of the denominator, 7+3i.

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

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

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

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

$$\frac{\left(2+5i\right)\left(7+3i\right)}{58}$$

Multiply complex numbers 2+5i and 7+3i like you multiply binomials.

$$\frac{2\times 7+2\times \left(3i\right)+5i\times 7+5\times 3i^{2}}{58}$$

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

$$\frac{2\times 7+2\times \left(3i\right)+5i\times 7+5\times 3\left(-1\right)}{58}$$

Do the multiplications in 2\times 7+2\times \left(3i\right)+5i\times 7+5\times 3\left(-1\right).

$$\frac{14+6i+35i-15}{58}$$

Combine the real and imaginary parts in 14+6i+35i-15.

$$\frac{14-15+\left(6+35\right)i}{58}$$

Do the additions in 14-15+\left(6+35\right)i.

$$\frac{-1+41i}{58}$$

Divide -1+41i by 58 to get -\frac{1}{58}+\frac{41}{58}i.

$$-\frac{1}{58}+\frac{41}{58}i$$

Real Part

$$-\frac{1}{58} = -0.017241379310344827$$

Solution Steps

Multiply both numerator and denominator of \frac{2+5i}{7-3i} by the complex conjugate of the denominator, 7+3i.

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

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

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

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

$$Re(\frac{\left(2+5i\right)\left(7+3i\right)}{58})$$

Multiply complex numbers 2+5i and 7+3i like you multiply binomials.

$$Re(\frac{2\times 7+2\times \left(3i\right)+5i\times 7+5\times 3i^{2}}{58})$$

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

$$Re(\frac{2\times 7+2\times \left(3i\right)+5i\times 7+5\times 3\left(-1\right)}{58})$$

Do the multiplications in 2\times 7+2\times \left(3i\right)+5i\times 7+5\times 3\left(-1\right).

$$Re(\frac{14+6i+35i-15}{58})$$

Combine the real and imaginary parts in 14+6i+35i-15.

$$Re(\frac{14-15+\left(6+35\right)i}{58})$$

Do the additions in 14-15+\left(6+35\right)i.

$$Re(\frac{-1+41i}{58})$$

Divide -1+41i by 58 to get -\frac{1}{58}+\frac{41}{58}i.

$$Re(-\frac{1}{58}+\frac{41}{58}i)$$

The real part of -\frac{1}{58}+\frac{41}{58}i is -\frac{1}{58}.

$$-\frac{1}{58}$$