How do you simplify # [(3+2i)^ 3 / (-2+3i)^4] #?

Question

James Allen · Accepted Answer

# (3-2i)/13# # 3+2i=sqrt(3^2+2^2)e^{i phi}# #-2+3i=sqrt(3^2+2^2)e^{i( phi+pi/2)}# with #phi = arctan(2/3)# # [(3+2i)^ 3 / (-2+3i)^4] =((3+2i)/(-2+3i))^3/( (-2+3i))=e^{-i (3pi)/2}/(sqrt(3^2+2^2)e^{i( phi+pi/2)})# #=1/sqrt(3^2+2^2)e^{-i(phi+2pi)} = 1/sqrt(3^2+2^2)e^{-i phi} =# #sqrt(3^2+2^2)/(3^2+2^2)e^{-i phi} = (3-2i)/13#

Madelyn Anderson · Answer

undefined To simplify the expression [(3+2i)^3 / (-2+3i)^4], you can follow these steps:

1. Expand the expressions (3+2i)^3 and (-2+3i)^4.
2. Simplify the expanded expressions.
3. Divide the simplified expression of (3+2i)^3 by the simplified expression of (-2+3i)^4.

Here's a step-by-step guide:

1. Expand (3+2i)^3:
   (3+2i)^3 = (3+2i)(3+2i)(3+2i)
            = (9 + 6i + 6i + 4i^2)(3+2i)
            = (9 + 12i - 4)(3+2i)
            = (5 + 12i)(3+2i)
            = 15 + 10i + 36i + 24i^2
            = 15 + 46i - 24  (since i^2 = -1)
            = -9 + 46i

2. Expand (-2+3i)^4:
   (-2+3i)^4 = (-2+3i)(-2+3i)(-2+3i)(-2+3i)
             = (-2+3i)^2 * (-2+3i)^2
             = (4 - 12i + 9i^2) * (4 - 12i + 9i^2)
             = (4 - 12i - 9) * (4 - 12i - 9)  (since i^2 = -1)
             = (-5 - 12i) * (-5 - 12i)
             = 25 + 60i - 60i - 144i^2
             = 25 - 144(-1)
             = 25 + 144
             = 169

3. Divide (-9 + 46i) by 169:
   (-9 + 46i) / 169 = -9/169 + 46i/169
                    = -9/169 + (46/169)i

So, the simplified expression is:
   [(3+2i)^3 / (-2+3i)^4] = (-9/169) + (46/169)i