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.