Solve 4(3+2i)^5-4-2i/4-3i

$$4 ( 3 + 2 i ) ^ { 5 } - \frac { 4 - 2 i } { 4 - 3 i }$$

Evaluate

$$-\frac{59722}{25}+\frac{12196}{25}i=-2388.88+487.84i$$

Solution Steps

Calculate 3+2i to the power of 5 and get -597+122i.

$$4\left(-597+122i\right)-\frac{4-2i}{4-3i}$$

Multiply 4 and -597+122i to get -2388+488i.

$$-2388+488i-\frac{4-2i}{4-3i}$$

Multiply both numerator and denominator of \frac{4-2i}{4-3i} by the complex conjugate of the denominator, 4+3i.

$$-2388+488i-\frac{\left(4-2i\right)\left(4+3i\right)}{\left(4-3i\right)\left(4+3i\right)}$$

Do the multiplications in \frac{\left(4-2i\right)\left(4+3i\right)}{\left(4-3i\right)\left(4+3i\right)}.

$$-2388+488i-\frac{22+4i}{25}$$

Divide 22+4i by 25 to get \frac{22}{25}+\frac{4}{25}i.

$$-2388+488i+\left(-\frac{22}{25}-\frac{4}{25}i\right)$$

Add -2388+488i and -\frac{22}{25}-\frac{4}{25}i to get -\frac{59722}{25}+\frac{12196}{25}i.

$$-\frac{59722}{25}+\frac{12196}{25}i$$

Real Part

$$-\frac{59722}{25} = -2388\frac{22}{25} = -2388.88$$

Solution Steps

Calculate 3+2i to the power of 5 and get -597+122i.

$$Re(4\left(-597+122i\right)-\frac{4-2i}{4-3i})$$

Multiply 4 and -597+122i to get -2388+488i.

$$Re(-2388+488i-\frac{4-2i}{4-3i})$$

Multiply both numerator and denominator of \frac{4-2i}{4-3i} by the complex conjugate of the denominator, 4+3i.

$$Re(-2388+488i-\frac{\left(4-2i\right)\left(4+3i\right)}{\left(4-3i\right)\left(4+3i\right)})$$

Do the multiplications in \frac{\left(4-2i\right)\left(4+3i\right)}{\left(4-3i\right)\left(4+3i\right)}.

$$Re(-2388+488i-\frac{22+4i}{25})$$

Divide 22+4i by 25 to get \frac{22}{25}+\frac{4}{25}i.

$$Re(-2388+488i+\left(-\frac{22}{25}-\frac{4}{25}i\right))$$

Add -2388+488i and -\frac{22}{25}-\frac{4}{25}i to get -\frac{59722}{25}+\frac{12196}{25}i.

$$Re(-\frac{59722}{25}+\frac{12196}{25}i)$$

The real part of -\frac{59722}{25}+\frac{12196}{25}i is -\frac{59722}{25}.

$$-\frac{59722}{25}$$