# Convert all complex numbers to trigonometric form and then simplify the expression. Write the answer in standard form.? ((2+2i)^5 (-3+i)^3) /(sqrt 3+i)^10

May 14, 2018

$\frac{27 \sqrt{3}}{4} - i \frac{27}{4}$

#### Explanation:

So this took me so many attempts and here it goes:

Find the r values using this formula:

$r = \sqrt{{x}^{2} + {y}^{2}}$

Plug in:

$\frac{{\left[2 \sqrt{2} \left(\cos {45}^{o} + i \sin {45}^{o}\right)\right]}^{5} {\left[3 \sqrt{2} \left(\cos {135}^{o} + i \sin {135}^{o}\right)\right]}^{3}}{2 \left(\cos {30}^{o} + i \sin {30}^{o}\right)} ^ 10$

Power and Multiply:

${\left(2 \sqrt{2}\right)}^{5}$ see how I powered the $2 \sqrt{2}$ and you do the same with the rest

$\frac{{\left(2 \sqrt{2}\right)}^{5} \left(\cos 5 \times {45}^{o} + i \sin 5 \times {45}^{o}\right) {\left(3 \sqrt{2}\right)}^{3} \left(\cos 3 \times {135}^{o} + i \sin 3 \times {135}^{o}\right)}{{2}^{10} \left(\cos 10 \times + i \sin 10 \times 30\right)}$

Simplify:

$\frac{128 \sqrt{2} \left(\cos {225}^{o} + i \sin {225}^{o}\right) 54 \sqrt{2} \left(\cos {405}^{o} + i \sin {405}^{o}\right)}{1024 \left(\cos {300}^{o} + i \sin {300}^{o}\right)}$

$\frac{13824 \left(\cos {630}^{o} + i \sin {630}^{o}\right)}{1024 \left(\cos {300}^{o} + i \sin {300}^{o}\right)}$

Divide:

$\frac{27}{2}$ $\left(\cos {630}^{o} - {300}^{o} + i \sin {630}^{o} - {300}^{o}\right)$

Simplify:

$\frac{27}{2}$ $\left(\cos {330}^{o} + i \sin {330}^{o}\right)$