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Final Answer:
1. $-i$
2. $-i$
3. $1$
4. $-1$
5. $i$
6. $i\sqrt{243}$ or $i \cdot 3\sqrt{27}$ → simplified: $i \cdot 3 \cdot 3\sqrt{3} = 9i\sqrt{3}$, but since 243 = $81 \cdot 3 = 9^2 \cdot 3$, $\sqrt{-243} = i\sqrt{243} = i \cdot 9\sqrt{3} = 9i\sqrt{3}$
Wait — let’s double-check: the problem likely expects simplest radical form with integer under root if possible, but 243 = $81 \times 3$, so $\sqrt{243} = 9\sqrt{3}$, so $\sqrt{-243} = 9i\sqrt{3}$. However, many worksheets accept $i\sqrt{243}$, but standard simplification is $9i\sqrt{3}$.
But looking at problems 7 and 8, they use perfect squares:
7. $\sqrt{-49} = 7i$
8. $\sqrt{-567}$: 567 = $81 \times 7 = 9^2 \cdot 7$, so $\sqrt{-567} = 9i\sqrt{7}$

So correct simplified answers:
6. $9i\sqrt{3}$
7. $7i$
8. $9i\sqrt{7}$

But the multiple-choice format only appears in Q1–5; Q6–8 are open. Since the user only asked for final answer and no options given for 6–8, we give simplified exact forms.

However, rechecking instructions: “Provide only the final correct answer.” And the worksheet shows Q1–5 with choices, Q6–8 without. Since the task is to solve the problem accurately, and the student must fill in answers, we list all 8 answers clearly.

Let me finalize each:

1. $i^{47}$: cycle of 4: 47 ÷ 4 = 11 rem 3 → $i^3 = -i$
2. $i^{21}$: 21 ÷ 4 = 5 rem 1 → $i^1 = i$? Wait! Mistake earlier.

Hold on — critical correction:

Powers of $i$ repeat every 4:
- $i^1 = i$
- $i^2 = -1$
- $i^3 = -i$
- $i^4 = 1$
Then repeats.

So:
1. $i^{47}$: 47 mod 4 = 47 − 44 = 3 → $i^3 = -i$ ✔
2. $i^{21}$: 21 mod 4 = 1 → $i^1 = i$ → but option list includes $i$, so answer is $i$, not $-i$! Earlier I said $-i$ — that was wrong.
3. $i^{24}$: 24 mod 4 = 0 → $i^4 = 1$ → answer $1$
4. $i^2 = -1$
5. $i^9$: 9 mod 4 = 1 → $i^1 = i$

So corrected:
1. $-i$
2. $i$
3. $1$
4. $-1$
5. $i$

6. $\sqrt{-243} = \sqrt{-1 \cdot 243} = i\sqrt{243} = i\sqrt{81 \cdot 3} = i \cdot 9\sqrt{3} = 9i\sqrt{3}$
7. $\sqrt{-49} = i\sqrt{49} = 7i$
8. $\sqrt{-567} = i\sqrt{567} = i\sqrt{81 \cdot 7} = 9i\sqrt{7}$

Now output only final answers as requested.

Final Answer:
1. $-i$
2. $i$
3. $1$
4. $-1$
5. $i$
6. $9i\sqrt{3}$
7. $7i$
8. $9i\sqrt{7}$