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The image contains three math problems. Let’s go through each one and solve them.

**6. Find the smallest number that should be added to 29870 to make it a perfect square.**

To solve this, we need to find the next perfect square that is greater than 29870.

1. Find the square root of 29870:
   \[
   \sqrt{29870} \approx 172.88
   \]
2. The next integer greater than 172.88 is 173. So, calculate \( 173^2 \):
   \[
   173^2 = 29929
   \]
3. Now, subtract 29870 from 29929 to find the difference:
   \[
   29929 - 29870 = 59
   \]
Thus, the smallest number that should be added to 29870 to make it a perfect square is **59**.

---

**7. Find the greatest 5-digit number that is a perfect square.**

1. The largest 5-digit number is 99999. Find the square root of 99999:
   \[
   \sqrt{99999} \approx 316.22
   \]
2. The greatest integer less than or equal to 316.22 is 316. Now, calculate \( 316^2 \):
   \[
   316^2 = 99856
   \]
Thus, the greatest 5-digit number that is a perfect square is **99856**.

---

**8. Find the smallest 5-digit number that is a perfect square.**

1. The smallest 5-digit number is 10000. Find the square root of 10000:
   \[
   \sqrt{10000} = 100
   \]
2. Since \( 100^2 = 10000 \), the smallest 5-digit number that is a perfect square is **10000**.

---

Would you like more details on any of these solutions? Here are 5 related questions to expand your understanding:

1. How can we estimate square roots of non-perfect square numbers more accurately?
2. What is the difference between perfect squares and perfect cubes?
3. How can we find perfect squares of larger numbers without using a calculator?
4. Can you find the next perfect square after 99856?
5. What methods can be used to determine if a number is a perfect square without calculating square roots?

**Tip:** Memorizing squares of integers up to at least 30 can help quickly recognize perfect squares in problem-solving.

6. Find the smallest number that should be added to 29870 to make it a perfect square.
7. Find the greatest 5-digit number that is a perfect square.
8. Find the smallest 5-digit number that is a perfect square.

Explore how to find the smallest number to add to 29870 to make it a perfect square, the greatest 5-digit perfect square, and the smallest 5-digit perfect square. This problem covers key mathematical concepts, including square roots, perfect squares, and approximation methods.

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