1. State the problem: Evaluate $\left(\sqrt{4} + 3i\right)^5$. 2. Simplify inside the parenthesis: $\sqrt{4} = 2$, so the expression becomes $\left(2 + 3i\right)^5$. 3. Use the binomial theorem or convert to polar form for easier exponentiation. 3a. Convert $2 + 3i$ to polar form: - Calculate modulus $r = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}$. - Calculate argument $\theta = \tan^{-1}\left(\frac{3}{2}\right)$. 3b. Express in polar form: $2 + 3i = \sqrt{13} (\cos\theta + i\sin\theta)$. 4. Apply De Moivre's theorem: $$\left(\sqrt{13}\right)^5 \left(\cos(5\theta) + i \sin(5\theta)\right) = 13^{\frac{5}{2}} (\cos(5\theta) + i \sin(5\theta))$$ 5. Calculate $r^5 = 13^{2.5} = (13^2) \times \sqrt{13} = 169 \times \sqrt{13}$. 6. Calculate $5\theta = 5 \times \tan^{-1}(\frac{3}{2})$. 7. Use decimal approximations: - $\tan^{-1}(1.5) \approx 0.9828$ radians - $5\theta \approx 4.914$ radians 8. Calculate cosine and sine: - $\cos(4.914) \approx 0.193$ - $\sin(4.914) \approx -0.981$ 9. Compute the final result: $$169 \times \sqrt{13} \times (0.193 - 0.981i)$$ - Note: $\sqrt{13} \approx 3.605$ - So, modulus $\approx 169 \times 3.605 = 609.345$ 10. Final value: - Real part: $609.345 \times 0.193 \approx 117.60$ - Imaginary part: $609.345 \times (-0.981) \approx -598.17$ Therefore, $$\left(2 + 3i\right)^5 \approx 117.6 - 598.17i$$.