Let's solve each problem step by step. The goal is to
estimate each square root by finding the two
whole numbers between which the square root lies.
We do this by identifying perfect squares near the given number and using the fact that:
> If $ a^2 < x < b^2 $, then $ a < \sqrt{x} < b $
---
1. $ \sqrt{5} $
Perfect squares:
- $ 2^2 = 4 $
- $ 3^2 = 9 $
Since $ 4 < 5 < 9 $, then $ \sqrt{5} $ is between
2 and 3
✔ Answer: 2 and 3
---
2. $ \sqrt{12} $
Perfect squares:
- $ 3^2 = 9 $
- $ 4^2 = 16 $
$ 9 < 12 < 16 $ → $ \sqrt{12} $ is between
3 and 4
✔ Answer: 3 and 4
---
3. $ \sqrt{34} $
Perfect squares:
- $ 5^2 = 25 $
- $ 6^2 = 36 $
$ 25 < 34 < 36 $ → $ \sqrt{34} $ is between
5 and 6
✔ Answer: 5 and 6
---
4. $ \sqrt{23} $
Perfect squares:
- $ 4^2 = 16 $
- $ 5^2 = 25 $
$ 16 < 23 < 25 $ → $ \sqrt{23} $ is between
4 and 5
✔ Answer: 4 and 5
---
5. $ \sqrt{101} $
Perfect squares:
- $ 10^2 = 100 $
- $ 11^2 = 121 $
$ 100 < 101 < 121 $ → $ \sqrt{101} $ is between
10 and 11
✔ Answer: 10 and 11
---
6. $ \sqrt{68} $
Perfect squares:
- $ 8^2 = 64 $
- $ 9^2 = 81 $
$ 64 < 68 < 81 $ → $ \sqrt{68} $ is between
8 and 9
✔ Answer: 8 and 9
---
7. $ \sqrt{41} $
Perfect squares:
- $ 6^2 = 36 $
- $ 7^2 = 49 $
$ 36 < 41 < 49 $ → $ \sqrt{41} $ is between
6 and 7
✔ Answer: 6 and 7
---
8. $ \sqrt{96} $
Perfect squares:
- $ 9^2 = 81 $
- $ 10^2 = 100 $
$ 81 < 96 < 100 $ → $ \sqrt{96} $ is between
9 and 10
✔ Answer: 9 and 10
---
9. $ \sqrt{125} $
Perfect squares:
- $ 11^2 = 121 $
- $ 12^2 = 144 $
$ 121 < 125 < 144 $ → $ \sqrt{125} $ is between
11 and 12
✔ Answer: 11 and 12
---
10. $ \sqrt{116} $
Perfect squares:
- $ 10^2 = 100 $
- $ 11^2 = 121 $
$ 100 < 116 < 121 $ → $ \sqrt{116} $ is between
10 and 11
✔ Answer: 10 and 11
---
11. $ \sqrt{77} $
Perfect squares:
- $ 8^2 = 64 $
- $ 9^2 = 81 $
$ 64 < 77 < 81 $ → $ \sqrt{77} $ is between
8 and 9
✔ Answer: 8 and 9
---
12. $ \sqrt{142} $
Perfect squares:
- $ 11^2 = 121 $
- $ 12^2 = 144 $
$ 121 < 142 < 144 $ → $ \sqrt{142} $ is between
11 and 12
✔ Answer: 11 and 12
---
✔ Final Answers:
| Problem | Answer |
|--------|--------|
| 1. $ \sqrt{5} $ | between
2 and 3 |
| 2. $ \sqrt{12} $ | between
3 and 4 |
| 3. $ \sqrt{34} $ | between
5 and 6 |
| 4. $ \sqrt{23} $ | between
4 and 5 |
| 5. $ \sqrt{101} $ | between
10 and 11 |
| 6. $ \sqrt{68} $ | between
8 and 9 |
| 7. $ \sqrt{41} $ | between
6 and 7 |
| 8. $ \sqrt{96} $ | between
9 and 10 |
| 9. $ \sqrt{125} $ | between
11 and 12 |
| 10. $ \sqrt{116} $ | between
10 and 11 |
| 11. $ \sqrt{77} $ | between
8 and 9 |
| 12. $ \sqrt{142} $ | between
11 and 12 |
---
💡
Tip: To estimate square roots, always think of perfect squares near the number. This method helps you approximate without a calculator!
Parent Tip: Review the logic above to help your child master the concept of estimating irrational square roots worksheet.
