Solve left(3-2iright)^3+1-i^21/-1+i^19-5left(2+iright)/2-i-left(2+iright)left(2-iright)

$${ \left(3-2i \right) }^{ 3 } + \frac{ 1- { i }^{ 21 } }{ -1+ { i }^{ 19 } } - \frac{ 5 \left( 2+i \right) }{ 2-i } - \left( 2+i \right) \left( 2-i \right)$$

Evaluate

$$-17-49i$$

Solution Steps

Calculate 3-2i to the power of 3 and get -9-46i.

$$-9-46i+\frac{1-i^{21}}{-1+i^{19}}-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right)$$

Calculate i to the power of 21 and get i.

$$-9-46i+\frac{1-i}{-1+i^{19}}-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right)$$

Calculate i to the power of 19 and get -i.

$$-9-46i+\frac{1-i}{-1-i}-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right)$$

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

$$-9-46i+\frac{\left(1-i\right)\left(-1+i\right)}{\left(-1-i\right)\left(-1+i\right)}-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right)$$

Do the multiplications in \frac{\left(1-i\right)\left(-1+i\right)}{\left(-1-i\right)\left(-1+i\right)}.

$$-9-46i+\frac{2i}{2}-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right)$$

Divide 2i by 2 to get i.

$$-9-46i+i-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right)$$

Do the additions in -9-46i+i.

$$-\frac{5\left(2+i\right)}{2-i}-9-45i-\left(2+i\right)\left(2-i\right)$$

Multiply 2+i and 2-i to get 5.

$$-\frac{5\left(2+i\right)}{2-i}-9-45i-5$$

Do the additions.

$$-\frac{5\left(2+i\right)}{2-i}-14-45i$$

Multiply 5 and 2+i to get 10+5i.

$$-\frac{10+5i}{2-i}-14-45i$$

Multiply both numerator and denominator of \frac{10+5i}{2-i} by the complex conjugate of the denominator, 2+i.

$$-\frac{\left(10+5i\right)\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}-14-45i$$

Do the multiplications in \frac{\left(10+5i\right)\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}.

$$-\frac{15+20i}{5}-14-45i$$

Divide 15+20i by 5 to get 3+4i.

$$-3-4i-14-45i$$

Do the additions.

$$-17-49i$$

Real Part

$$-17$$

Solution Steps

Calculate 3-2i to the power of 3 and get -9-46i.

$$Re(-9-46i+\frac{1-i^{21}}{-1+i^{19}}-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right))$$

Calculate i to the power of 21 and get i.

$$Re(-9-46i+\frac{1-i}{-1+i^{19}}-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right))$$

Calculate i to the power of 19 and get -i.

$$Re(-9-46i+\frac{1-i}{-1-i}-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right))$$

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

$$Re(-9-46i+\frac{\left(1-i\right)\left(-1+i\right)}{\left(-1-i\right)\left(-1+i\right)}-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right))$$

Do the multiplications in \frac{\left(1-i\right)\left(-1+i\right)}{\left(-1-i\right)\left(-1+i\right)}.

$$Re(-9-46i+\frac{2i}{2}-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right))$$

Divide 2i by 2 to get i.

$$Re(-9-46i+i-\frac{5\left(2+i\right)}{2-i}-\left(2+i\right)\left(2-i\right))$$

Do the additions in -9-46i+i.

$$Re(-\frac{5\left(2+i\right)}{2-i}-9-45i-\left(2+i\right)\left(2-i\right))$$

Multiply 2+i and 2-i to get 5.

$$Re(-\frac{5\left(2+i\right)}{2-i}-9-45i-5)$$

Do the additions in -9-45i-5.

$$Re(-\frac{5\left(2+i\right)}{2-i}-14-45i)$$

Multiply 5 and 2+i to get 10+5i.

$$Re(-\frac{10+5i}{2-i}-14-45i)$$

Multiply both numerator and denominator of \frac{10+5i}{2-i} by the complex conjugate of the denominator, 2+i.

$$Re(-\frac{\left(10+5i\right)\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}-14-45i)$$

Do the multiplications in \frac{\left(10+5i\right)\left(2+i\right)}{\left(2-i\right)\left(2+i\right)}.

$$Re(-\frac{15+20i}{5}-14-45i)$$

Divide 15+20i by 5 to get 3+4i.

$$Re(-3-4i-14-45i)$$

Do the additions in -3-4i-14-45i.

$$Re(-17-49i)$$

The real part of -17-49i is -17.

$$-17$$