Square root estimation answers worksheet with filled-in blanks and calculations.

The image you've uploaded is a completed worksheet titled *"Answers"* that focuses on estimating square roots. The task asks students to use their square root estimation skills to determine between which two consecutive integers each given square root lies.

Let’s go through the logic and reasoning behind how these answers were determined, so you can understand the method used.

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🔍 How to Estimate Square Roots

To estimate √x:
1. Find two perfect squares such that:
$$
a^2 < x < b^2
$$
2. Then:
$$
a < \sqrt{x} < b
$$

So, we look for perfect squares just below and just above the number under the square root.

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✔ Let's verify a few examples:

#### 1. √200
- 14² = 196
- 15² = 225
- So, 196 < 200 < 225 → √200 is between 14 and 15

✔ Correct.

#### 2. √37
- 6² = 36
- 7² = 49
- 36 < 37 < 49 → √37 is between 6 and 7

✔ Correct.

#### 3. √31
- 5² = 25
- 6² = 36
- 25 < 31 < 36 → √31 is between 5 and 6

✔ Correct.

#### 4. √2
- 1² = 1
- 2² = 4
- 1 < 2 < 4 → √2 is between 1 and 2

✔ Correct.

#### 5. √134
- 11² = 121
- 12² = 144
- 121 < 134 < 144 → √134 is between 11 and 12

✔ Correct.

#### 6. √350
- 18² = 324
- 19² = 361
- 324 < 350 < 361 → √350 is between 18 and 19

✔ Correct.

#### 7. √8
- 2² = 4
- 3² = 9
- 4 < 8 < 9 → √8 is between 2 and 3

✔ Correct.

#### 8. √500
- 22² = 484
- 23² = 529
- 484 < 500 < 529 → √500 is between 22 and 23

✔ Correct.

#### 9. √80
- 8² = 64
- 9² = 81
- 64 < 80 < 81 → √80 is between 8 and 9

✔ Correct.

#### 10. √60
- 7² = 49
- 8² = 64
- 49 < 60 < 64 → √60 is between 7 and 8

✔ Correct.

All other entries follow the same logic.

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🧠 Summary of the Method:

For any number x, find the nearest perfect squares:
- Identify n² < x < (n+1)²
- Then: √x is between n and n+1

This technique helps estimate irrational square roots without a calculator.

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✔ Final Answer:

All the filled-in answers in the image are correct and consistent with proper square root estimation.

Here’s a quick list of all the answers:

| √x | Between |
|-------|--------|
| √200 | 14 and 15 |
| √37 | 6 and 7 |
| √31 | 5 and 6 |
| √2 | 1 and 2 |
| √134 | 11 and 12 |
| √350 | 18 and 19 |
| √8 | 2 and 3 |
| √500 | 22 and 23 |
| √80 | 8 and 9 |
| √60 | 7 and 8 |
| √111 | 10 and 11 |
| √230 | 15 and 16 |
| √320 | 17 and 18 |
| √190 | 13 and 14 |
| √3 | 1 and 2 |
| √5 | 2 and 3 |
| √150 | 12 and 13 |
| √115 | 10 and 11 |
| √70 | 8 and 9 |
| √90 | 9 and 10 |

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💡 Tip:

You can improve your estimation by checking how close the number is to the nearest perfect square. For example:
- √80 is very close to 81 → √80 ≈ 8.9
- √190 is closer to 196 (14²) than 169 (13²), but still between 13 and 14.

But for this worksheet, only the integer bounds are required.

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✔ Conclusion: The worksheet has been correctly completed. All answers are accurate based on square root estimation principles.