Estimating Square Roots worksheet with practice problems and example solution.

Let's solve each square root estimation problem step by step. The goal is to estimate the square root of a number by finding two consecutive whole numbers between which the square root lies.

We do this by identifying perfect squares close to the given number and using them as bounds.

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🔢 Step-by-step Method:

For any number $ x $, find two perfect squares such that:
$$
a^2 < x < b^2 \Rightarrow \sqrt{x} \text{ is between } a \text{ and } b
$$

Let’s go through each one:

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1. $ \sqrt{200} $
$ 14^2 = 196 $, $ 15^2 = 225 $ → $ 196 < 200 < 225 $
✔ Between 14 and 15

2. $ \sqrt{37} $
$ 6^2 = 36 $, $ 7^2 = 49 $ → $ 36 < 37 < 49 $
✔ Between 6 and 7

3. $ \sqrt{31} $
$ 5^2 = 25 $, $ 6^2 = 36 $ → $ 25 < 31 < 36 $
✔ Between 5 and 6

4. $ \sqrt{2} $
$ 1^2 = 1 $, $ 2^2 = 4 $ → $ 1 < 2 < 4 $
✔ Between 1 and 2

5. $ \sqrt{134} $
$ 11^2 = 121 $, $ 12^2 = 144 $ → $ 121 < 134 < 144 $
✔ Between 11 and 12

6. $ \sqrt{350} $
$ 18^2 = 324 $, $ 19^2 = 361 $ → $ 324 < 350 < 361 $
✔ Between 18 and 19

7. $ \sqrt{8} $
$ 2^2 = 4 $, $ 3^2 = 9 $ → $ 4 < 8 < 9 $
✔ Between 2 and 3

8. $ \sqrt{500} $
$ 22^2 = 484 $, $ 23^2 = 529 $ → $ 484 < 500 < 529 $
✔ Between 22 and 23

9. $ \sqrt{80} $
$ 8^2 = 64 $, $ 9^2 = 81 $ → $ 64 < 80 < 81 $
✔ Between 8 and 9

10. $ \sqrt{60} $
$ 7^2 = 49 $, $ 8^2 = 64 $ → $ 49 < 60 < 64 $
✔ Between 7 and 8

11. $ \sqrt{111} $
$ 10^2 = 100 $, $ 11^2 = 121 $ → $ 100 < 111 < 121 $
✔ Between 10 and 11

12. $ \sqrt{230} $
$ 15^2 = 225 $, $ 16^2 = 256 $ → $ 225 < 230 < 256 $
✔ Between 15 and 16

13. $ \sqrt{320} $
$ 17^2 = 289 $, $ 18^2 = 324 $ → $ 289 < 320 < 324 $
✔ Between 17 and 18

14. $ \sqrt{190} $
$ 13^2 = 169 $, $ 14^2 = 196 $ → $ 169 < 190 < 196 $
✔ Between 13 and 14

15. $ \sqrt{3} $
$ 1^2 = 1 $, $ 2^2 = 4 $ → $ 1 < 3 < 4 $
✔ Between 1 and 2

16. $ \sqrt{5} $
$ 2^2 = 4 $, $ 3^2 = 9 $ → $ 4 < 5 < 9 $
✔ Between 2 and 3

17. $ \sqrt{150} $
$ 12^2 = 144 $, $ 13^2 = 169 $ → $ 144 < 150 < 169 $
✔ Between 12 and 13

18. $ \sqrt{115} $
$ 10^2 = 100 $, $ 11^2 = 121 $ → $ 100 < 115 < 121 $
✔ Between 10 and 11

19. $ \sqrt{70} $
$ 8^2 = 64 $, $ 9^2 = 81 $ → $ 64 < 70 < 81 $
✔ Between 8 and 9

20. $ \sqrt{90} $
$ 9^2 = 81 $, $ 10^2 = 100 $ → $ 81 < 90 < 100 $
✔ Between 9 and 10

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✔ Final Answers (Fill in the blanks):

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

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📝 Explanation Summary:

To estimate square roots:
- Find the perfect squares just below and above the number.
- Take their square roots — those are your bounds.
- For example: $ \sqrt{37} $: since $ 36 < 37 < 49 $, then $ 6 < \sqrt{37} < 7 $

This method gives a good approximation without needing a calculator.

✔ All answers are now filled in!