The Complex Power from a source V = 200cos(wt) is 200 + j150. The impedance across V is given by what value?
When one writes \$V=200\cos(\omega t)\$, \$V\$ is the instantaneous voltage, and 200 is the peak of that voltage. To calculate power you need to use the rms value the voltage, which for a sine wave is \$\frac{1}{\sqrt{2}} = 0.707\$ times the peak voltage.
The failure to use rms value is why the answer you arrived at is twice as large as the correct answer.
The complex power \$S\$ is given by:
\$S = V_{rms}\cdot I_{rms}^*\$
so
\$I_{rms}= \frac{S^*}{V_{rms}^*}\$
The impedance is given by
\$Z = \frac{V}{I} = \frac{V_{rms}}{I_{rms}} = \frac{V_{rms}V_{rms}^*}{S^*}\$
In the given problem,
\$Z = \frac{(\frac{ 200}{\sqrt{2}})^2}{200-j150}= \frac{20000}{200^2+150^2}(200+j150)=\frac{20000}{62500}(200+j150) = 64+j48\$