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Let's solve each cube root step by step.

We are asked to find the cube root of each number. The cube root of a number $ x $ is a number $ y $ such that:

$$
y^3 = x
$$

So we need to find a number that, when multiplied by itself three times, gives the original number.

---

1. $ \sqrt[3]{729} $

Let’s try some numbers:
- $ 8^3 = 512 $
- $ 9^3 = 729 $

✔ So, $ \sqrt[3]{729} = 9 $

---

2. $ \sqrt[3]{1000} $

Try:
- $ 10^3 = 1000 $

✔ So, $ \sqrt[3]{1000} = 10 $

---

3. $ \sqrt[3]{1} $

Any number to the power of 3: $ 1^3 = 1 $

✔ So, $ \sqrt[3]{1} = 1 $

---

4. $ \sqrt[3]{2197} $

Let’s estimate:
- $ 10^3 = 1000 $
- $ 12^3 = 1728 $
- $ 13^3 = 13 \times 13 \times 13 = 169 \times 13 $

Calculate:
- $ 169 \times 13 = (170 - 1) \times 13 = 170 \times 13 - 13 = 2110 - 13 = 2097 $
- $ 14^3 = 14 \times 14 \times 14 = 196 \times 14 = 2744 $ → too big

Wait, $ 13^3 = 2197 $? Let's check:
- $ 13^2 = 169 $
- $ 169 \times 13 = ? $

Break it down:
- $ 169 \times 10 = 1690 $
- $ 169 \times 3 = 507 $
- Total: $ 1690 + 507 = 2197 $

✔ So, $ \sqrt[3]{2197} = 13 $

---

5. $ \sqrt[3]{64} $

Try:
- $ 4^3 = 4 \times 4 \times 4 = 16 \times 4 = 64 $

✔ So, $ \sqrt[3]{64} = 4 $

---

6. $ \sqrt[3]{27} $

- $ 3^3 = 27 $

✔ So, $ \sqrt[3]{27} = 3 $

---

7. $ \sqrt[3]{512} $

Try:
- $ 8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512 $

✔ So, $ \sqrt[3]{512} = 8 $

---

8. $ \sqrt[3]{216} $

Try:
- $ 6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216 $

✔ So, $ \sqrt[3]{216} = 6 $

---

9. $ \sqrt[3]{125} $

- $ 5^3 = 5 \times 5 \times 5 = 25 \times 5 = 125 $

✔ So, $ \sqrt[3]{125} = 5 $

---

10. $ \sqrt[3]{343} $

Try:
- $ 7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343 $

✔ So, $ \sqrt[3]{343} = 7 $

---

✔ Final Answers:

$$
\begin{align*}
\sqrt[3]{729} &= 9 \\
\sqrt[3]{1000} &= 10 \\
\sqrt[3]{1} &= 1 \\
\sqrt[3]{2197} &= 13 \\
\sqrt[3]{64} &= 4 \\
\sqrt[3]{27} &= 3 \\
\sqrt[3]{512} &= 8 \\
\sqrt[3]{216} &= 6 \\
\sqrt[3]{125} &= 5 \\
\sqrt[3]{343} &= 7 \\
\end{align*}
$$

These are all perfect cubes, so their cube roots are whole numbers.