Solve (-1+14/2i/8-3i)

$$( \frac { - 1 + \frac { 14 } { 2 } i } { 8 - 3 i } )$$

Evaluate

$$-\frac{29}{73}+\frac{53}{73}i\approx -0.397260274+0.726027397i$$

Solution Steps

Divide 14 by 2 to get 7.

$$\frac{-1+7i}{8-3i}$$

Multiply both numerator and denominator by the complex conjugate of the denominator, 8+3i.

$$\frac{\left(-1+7i\right)\left(8+3i\right)}{\left(8-3i\right)\left(8+3i\right)}$$

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

$$\frac{\left(-1+7i\right)\left(8+3i\right)}{8^{2}-3^{2}i^{2}}$$

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

$$\frac{\left(-1+7i\right)\left(8+3i\right)}{73}$$

Multiply complex numbers -1+7i and 8+3i like you multiply binomials.

$$\frac{-8-3i+7i\times 8+7\times 3i^{2}}{73}$$

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

$$\frac{-8-3i+7i\times 8+7\times 3\left(-1\right)}{73}$$

Do the multiplications in -8-3i+7i\times 8+7\times 3\left(-1\right).

$$\frac{-8-3i+56i-21}{73}$$

Combine the real and imaginary parts in -8-3i+56i-21.

$$\frac{-8-21+\left(-3+56\right)i}{73}$$

Do the additions in -8-21+\left(-3+56\right)i.

$$\frac{-29+53i}{73}$$

Divide -29+53i by 73 to get -\frac{29}{73}+\frac{53}{73}i.

$$-\frac{29}{73}+\frac{53}{73}i$$

Real Part

$$-\frac{29}{73} = -0.3972602739726027$$

Solution Steps

Divide 14 by 2 to get 7.

$$Re(\frac{-1+7i}{8-3i})$$

Multiply both numerator and denominator of \frac{-1+7i}{8-3i} by the complex conjugate of the denominator, 8+3i.

$$Re(\frac{\left(-1+7i\right)\left(8+3i\right)}{\left(8-3i\right)\left(8+3i\right)})$$

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

$$Re(\frac{\left(-1+7i\right)\left(8+3i\right)}{8^{2}-3^{2}i^{2}})$$

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

$$Re(\frac{\left(-1+7i\right)\left(8+3i\right)}{73})$$

Multiply complex numbers -1+7i and 8+3i like you multiply binomials.

$$Re(\frac{-8-3i+7i\times 8+7\times 3i^{2}}{73})$$

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

$$Re(\frac{-8-3i+7i\times 8+7\times 3\left(-1\right)}{73})$$

Do the multiplications in -8-3i+7i\times 8+7\times 3\left(-1\right).

$$Re(\frac{-8-3i+56i-21}{73})$$

Combine the real and imaginary parts in -8-3i+56i-21.

$$Re(\frac{-8-21+\left(-3+56\right)i}{73})$$

Do the additions in -8-21+\left(-3+56\right)i.

$$Re(\frac{-29+53i}{73})$$

Divide -29+53i by 73 to get -\frac{29}{73}+\frac{53}{73}i.

$$Re(-\frac{29}{73}+\frac{53}{73}i)$$

The real part of -\frac{29}{73}+\frac{53}{73}i is -\frac{29}{73}.

$$-\frac{29}{73}$$