The exercise sheet you uploaded covers the topic of simplifying imaginary numbers, particularly complex numbers of the form $(x + yi)^n$. The sheet includes several problems related to raising complex numbers to powers.

Problem Set:

1. $(1 + i)^5$
2. $(3 - 4i)^3$
3. $(-6 - 8i)^{10}$
4. $(3 - 4i)^3$ (repeated)
5. $(5 + 12i)^2$

The instruction on the sheet outlines that to solve such problems, you can use polar form:

• Magnitude (r): $r = \sqrt{x^2 + y^2}$
• Angle (A): $\tan(A) = \frac{y}{x}$
• The complex number can then be written in polar form as $r(\cos A + i\sin A)$, and powers can be raised using De Moivre's Theorem: $(r (\cos A + i \sin A))^n = r^n (\cos(nA) + i\sin(nA))$

Would you like me to solve these problems step by step for you, or do you have a specific one you'd like explained?

Expansion Questions:

1. How do you find the polar form of a complex number?
2. What is De Moivre’s Theorem, and how is it applied?
3. Can you explain how to compute powers of imaginary numbers without using polar form?
4. Why is it important to know both the rectangular and polar forms of a complex number?
5. How does the angle $A$ affect the outcome of the power operation?

Tip:

When working with imaginary numbers, converting them into polar form simplifies the multiplication and exponentiation processes by working with magnitudes and angles rather than real and imaginary components directly.