1. **Problem:** Calculate the following complex expressions: i. $(1 - 2i)^4$ ii. $\frac{2+i}{2-i}$ 2. **Formula and rules:** - To raise a complex number to a power, use binomial expansion or convert to polar form. - To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator. 3. **Solution for (1 - 2i)^4:** - First, find $(1 - 2i)^2$: $$ (1 - 2i)^2 = 1^2 - 2 \times 1 \times 2i + (2i)^2 = 1 - 4i + 4i^2 $$ Since $i^2 = -1$, this becomes: $$ 1 - 4i + 4(-1) = 1 - 4i - 4 = -3 - 4i $$ - Now square the result to get the fourth power: $$ (1 - 2i)^4 = (-3 - 4i)^2 = (-3)^2 + 2 \times (-3) \times (-4i) + (-4i)^2 = 9 + 24i + 16i^2 $$ Again, $i^2 = -1$, so: $$ 9 + 24i + 16(-1) = 9 + 24i - 16 = -7 + 24i $$ 4. **Solution for $\frac{2+i}{2-i}$:** - Multiply numerator and denominator by the conjugate of the denominator $2+i$: $$ \frac{2+i}{2-i} \times \frac{2+i}{2+i} = \frac{(2+i)^2}{(2)^2 - (i)^2} $$ - Calculate numerator: $$ (2+i)^2 = 2^2 + 2 \times 2 \times i + i^2 = 4 + 4i + (-1) = 3 + 4i $$ - Calculate denominator: $$ 2^2 - i^2 = 4 - (-1) = 5 $$ - So the expression is: $$ \frac{3 + 4i}{5} = \frac{3}{5} + \frac{4}{5}i $$ **Final answers:** - $(1 - 2i)^4 = -7 + 24i$ - $\frac{2+i}{2-i} = \frac{3}{5} + \frac{4}{5}i$