1. **State the problem:** Simplify the following expressions involving powers of the imaginary unit $i$: i) $i^9$ ii) $i^{14}$ iii) $(-i)^{19}$ iv) $\left(21^2 - \right)$ (Note: The last expression seems incomplete or unclear; assuming it is a typo or incomplete, we will focus on the first three.) 2. **Recall the key property of $i$:** The imaginary unit $i$ satisfies $i^2 = -1$. Powers of $i$ cycle every 4 steps: $$i^1 = i$$ $$i^2 = -1$$ $$i^3 = -i$$ $$i^4 = 1$$ Then the cycle repeats: $i^{n+4} = i^n$. 3. **Simplify each expression:** i) $i^9$ Since powers of $i$ repeat every 4, find $9 \mod 4$: $$9 \div 4 = 2 \text{ remainder } 1$$ So, $i^9 = i^{4\cdot 2 + 1} = i^1 = i$. ii) $i^{14}$ Find $14 \mod 4$: $$14 \div 4 = 3 \text{ remainder } 2$$ So, $i^{14} = i^{4\cdot 3 + 2} = i^2 = -1$. iii) $(-i)^{19}$ First, rewrite $-i$ as $-1 \times i$. So, $$(-i)^{19} = (-1)^{19} \times i^{19}$$ Since $(-1)^{19} = -1$ (because 19 is odd), and Find $19 \mod 4$: $$19 \div 4 = 4 \text{ remainder } 3$$ So, $i^{19} = i^3 = -i$. Therefore, $$(-i)^{19} = -1 \times (-i) = i$$. 4. **Summary of results:** i) $i^9 = i$ ii) $i^{14} = -1$ iii) $(-i)^{19} = i$ 5. **Note:** The fourth expression is unclear and cannot be simplified as given. **Final answers:** $$i^9 = i$$ $$i^{14} = -1$$ $$(-i)^{19} = i$$