Estimating Square Roots worksheet with ten problems requiring students to find the approximate square roots of numbers like √31, √53, and √-200, determining the range and closest integer value.

Let's solve each problem step by step. This worksheet asks you to estimate square roots and determine:

1. Between which two consecutive integers the square root falls.
2. Which integer it is closest to.

We’ll go through each one carefully, using perfect squares as reference points.

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Key Concept:

To estimate √n:
- Find two perfect squares between which n lies.
- Then the square root of n lies between the square roots of those perfect squares.
- Determine which perfect square is closer to n to find the nearest integer.

Also note:
- The square root of a negative number is not real (it’s imaginary), so we need to be careful with negative numbers.

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Problem 1: √31

- Perfect squares near 31:
- 25 = 5²
- 36 = 6²
- So √31 is between 5 and 6
- 31 is closer to 36 than to 25 → so √31 is closer to 6

✔ Answer:
Falls between 5 and 6, but it’s closest to 6

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Problem 2: √53

- Perfect squares:
- 49 = 7²
- 64 = 8²
- √53 is between 7 and 8
- 53 – 49 = 4; 64 – 53 = 11 → closer to 49 → closer to 7

✔ Answer:
Falls between 7 and 8, but it’s closest to 7

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Problem 3: √11

- Perfect squares:
- 9 = 3²
- 16 = 4²
- √11 is between 3 and 4
- 11 – 9 = 2; 16 – 11 = 5 → closer to 9 → closer to 3

✔ Answer:
Falls between 3 and 4, but it’s closest to 3

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Problem 4: √130

- Perfect squares:
- 121 = 11²
- 144 = 12²
- √130 is between 11 and 12
- 130 – 121 = 9; 144 – 130 = 14 → closer to 121 → closer to 11

✔ Answer:
Falls between 11 and 12, but it’s closest to 11

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Problem 5: √215

- Perfect squares:
- 196 = 14²
- 225 = 15²
- √215 is between 14 and 15
- 215 – 196 = 19; 225 – 215 = 10 → closer to 225 → closer to 15

✔ Answer:
Falls between 14 and 15, but it’s closest to 15

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Problem 6: √278

- Perfect squares:
- 256 = 16²
- 289 = 17²
- √278 is between 16 and 17
- 278 – 256 = 22; 289 – 278 = 11 → closer to 289 → closer to 17

✔ Answer:
Falls between 16 and 17, but it’s closest to 17

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Problem 7: √(-175)

⚠️ Important: The square root of a negative number is not a real number. It's an imaginary number.

But the worksheet says: *"Your answers must be between -20 and 20"*, and asks for where it "falls between".

However, since √(-175) is not real, we cannot place it on the real number line.

But perhaps the question meant to ask about -√175? Let’s check.

Wait — it says: "The square root of −√175"?

No — actually, it says: "The square root of −175", which is √(-175)

That’s not real.

But maybe it's a typo, or they want us to interpret it differently.

Alternatively, perhaps they mean: −√175, i.e., the negative square root of 175.

Let’s assume that it's asking for −√175, because otherwise the answer isn't real.

So let’s compute −√175.

First, √175:

- 169 = 13²
- 196 = 14²
- So √175 is between 13 and 14
- 175 – 169 = 6; 196 – 175 = 21 → closer to 169 → √175 ≈ 13.2 → so −√175 ≈ −13.2

So −√175 is between −14 and −13, and closer to −13

✔ Answer:
Falls between −14 and −13, but it’s closest to −13

> Note: This assumes the expression means −√175, not √(−175). Because √(−175) is imaginary.

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Problem 8: √87

- Perfect squares:
- 81 = 9²
- 100 = 10²
- √87 is between 9 and 10
- 87 – 81 = 6; 100 – 87 = 13 → closer to 81 → closer to 9

✔ Answer:
Falls between 9 and 10, but it’s closest to 9

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Problem 9: √(−21)

Again, this is not a real number — √(−21) is imaginary.

But if it's meant to be −√21, then:

- √21:
- 16 = 4²
- 25 = 5²
- √21 between 4 and 5
- 21 – 16 = 5; 25 – 21 = 4 → closer to 25 → √21 ≈ 4.58 → −√21 ≈ −4.58

So −√21 is between −5 and −4, and closer to −5 (since 4.58 is closer to 5 than to 4)

✔ Answer:
Falls between −5 and −4, but it’s closest to −5

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Problem 10: √(−200)

Same issue — square root of a negative number → not real

But if it's −√200, then:

- √200:
- 196 = 14²
- 225 = 15²
- √200 between 14 and 15
- 200 – 196 = 4; 225 – 200 = 25 → closer to 196 → √200 ≈ 14.14 → −√200 ≈ −14.14

So −√200 is between −15 and −14, and closer to −14

✔ Answer:
Falls between −15 and −14, but it’s closest to −14

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## ✔ Final Answers:

| Problem | Falls Between | Closest To |
|--------|---------------|------------|
| 1. √31 | 5 and 6 | 6 |
| 2. √53 | 7 and 8 | 7 |
| 3. √11 | 3 and 4 | 3 |
| 4. √130 | 11 and 12 | 11 |
| 5. √215 | 14 and 15 | 15 |
| 6. √278 | 16 and 17 | 17 |
| 7. √(-175) → assumed −√175 | −14 and −13 | −13 |
| 8. √87 | 9 and 10 | 9 |
| 9. √(-21) → assumed −√21 | −5 and −4 | −5 |
| 10. √(-200) → assumed −√200 | −15 and −14 | −14 |

> ⚠️ Note: Problems 7, 9, and 10 involve square roots of negative numbers, which are not real. However, based on context and the instruction that answers must be between -20 and 20, it's likely that the intended meaning was negative square roots (e.g., −√175, not √(−175)). We’ve interpreted them accordingly.

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📌 Summary of How to Estimate Square Roots:

1. Find perfect squares around the number.
2. Take their square roots → gives bounds.
3. See which perfect square the number is closer to → determines closest integer.

For negative values, if the expression is −√x, then:
- First estimate √x
- Then take the negative
- Place it on the number line accordingly

Let me know if you'd like a visual explanation or help with any specific one!