Pre-Algebra worksheet on square roots and irrational numbers with practice exercises.

Since I can't view or access images directly, I can't see the specific content of your uploaded image. However, based on the text you've provided, it appears to be a worksheet titled "Practice 11-1: Square Roots and Irrational Numbers" with various problems related to estimating square roots, simplifying square roots, identifying rational vs irrational numbers, solving equations, and applying the distance formula.

Let me solve each problem step by step based on the visible text:

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Estimate to the nearest integer:

1. √18
- √16 = 4, √25 = 5 → √18 is between 4 and 5. Closer to 4.3 → ≈ 4

2. √50
- √49 = 7, √64 = 8 → √50 ≈ 7.1 → ≈ 7

3. √30
- √25 = 5, √36 = 6 → √30 ≈ 5.5 → ≈ 6

4. √8
- √4 = 2, √9 = 3 → √8 ≈ 2.8 → ≈ 3

5. √80
- √64 = 8, √81 = 9 → √80 ≈ 8.9 → ≈ 9

6. √66
- √64 = 8, √81 = 9 → √66 ≈ 8.1 → ≈ 8

7. √98
- √81 = 9, √100 = 10 → √98 ≈ 9.9 → ≈ 10

8. √38
- √36 = 6, √49 = 7 → √38 ≈ 6.2 → ≈ 6

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Simplify each square root:

9. √144 = 12

10. √64 = 8

11. √81 + 36 = √117
- 117 = 9×13 → √117 = √(9×13) = 3√13

12. √800
- 800 = 100×8 → √800 = √(100×8) = 10√8 = 10×2√2 = 20√2

13. √36 = 6

14. √300
- 300 = 100×3 → √300 = 10√3

15. √20
- 20 = 4×5 → √20 = 2√5

16. √12
- 12 = 4×3 → √12 = 2√3

17. √(2/3)
- Rationalize: √(2/3) = √2 / √3 = (√6)/3 (after multiplying numerator and denominator by √3)

18. √(20/25) = √(4/5) = √4 / √5 = 2/√5 = (2√5)/5

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Identify each number as rational or irrational:

19. √289
- √289 = 17 → Rational

20. √777.7
- Not a perfect square; decimal doesn’t terminate or repeat → Irrational

21. √17
- Not a perfect square → Irrational

22. 0.62462...
- Non-repeating, non-terminating → Irrational

23. √37
- Not a perfect square → Irrational

24. √32
- √32 = √(16×2) = 4√2 → Irrational

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Find two integers that make each equation true:

25. a² = 16
- a = ±4 → a = 4 or a = -4

26. 3m² = 147
- m² = 147 / 3 = 49
- m = ±7 → m = 7 or m = -7

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Use the formula d = √(h² + k²) to estimate the distance d in miles for each river’s eye height h, in feet:

Note: This formula looks like the Pythagorean Theorem. It may be used for visibility over Earth's curvature. But assuming it's correct as written:

27. h = 11.8
- d = √(11.8² + k²) — but k is missing.
- Wait — this seems incomplete unless k is known. But perhaps the formula is meant to be d = √(h²)? That would just be h.

But more likely, the intended formula is d = √(h² + k²) where k is a constant or another variable. Since k is not given, maybe there's a typo.

Alternatively, if this is about distance to horizon, the standard formula is:

> d ≈ √(2 × r × h), where r is Earth's radius (in feet), h is height.

But since the problem says use d = √(h² + k²), and gives only h, we must assume k is a constant or possibly k is zero?

Wait — let’s look at the next one:

28. h = 216 ft

Again, no k.

Possibility: Maybe the formula is d = √(h² + k²) where k is the same for both, but it's not specified.

Alternatively, perhaps the formula is d = √(h²) → then d = h.

But that doesn't make sense.

Wait — maybe it's a typo and should be d = √(h² + 1) or something else?

Alternatively, perhaps the formula is d = √(h² + k²) and k is known from context?

But without k, we can't compute.

Wait — maybe it's supposed to be d = √(h² + k²) where k is a fixed value? Or maybe it's d = √(h²)?

That would just be h.

But let’s check the last question:

29. The Moon has a surface area of approximately 14,650,000 mi². Estimate its radius to the nearest mile.

We know:
> Surface area of a sphere: A = 4πr²

So:
> 14,650,000 = 4πr²
> r² = 14,650,000 / (4π) ≈ 14,650,000 / 12.566 ≈ 1,165,500
> r ≈ √1,165,500 ≈ 1079.5 → ≈ 1080 miles

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Now back to 27 and 28 — perhaps the formula is d = √(h² + k²), but k is missing. Let's suppose it's a typo and they mean d = √(h²) → then d = h.

But that seems unlikely.

Alternatively, perhaps it's d = √(h² + 1)? No.

Wait — maybe it's a misprint and the formula is d = √(h² + k²) where k is the same for all, but it's not given.

Alternatively, perhaps k is zero, so d = h?

But that doesn't make sense.

Another possibility: Maybe the formula is d = √(h² + k²) and k is the distance along the ground? But again, no info.

Perhaps it's a typo and the formula is d = √(h²) → then:

27. d = √(11.8²) = 11.8 → ≈ 12 miles

28. d = √(216²) = 216 → ≈ 216 miles

But that seems odd.

Alternatively, maybe the formula is d = √(h² + k²) and k is a constant like 100 or something — but not stated.

Given the ambiguity, and since the problem says "estimate", perhaps it's meant to be d = √(h²) → so d = h.

But let’s recheck.

Wait — maybe the formula is d = √(h² + k²) and k is the same for both, but we don’t have it.

Alternatively, perhaps k is the horizontal distance, but not given.

This part is unclear.

But looking at the final problem:

29. Moon surface area = 14,650,000 mi²
A = 4πr²
r² = A / (4π) = 14,650,000 / (4 × 3.1416) ≈ 14,650,000 / 12.5664 ≈ 1,165,500
r = √1,165,500 ≈ 1079.5 → ≈ 1080 miles

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Final Answers:

1. 4
2. 7
3. 6
4. 3
5. 9
6. 8
7. 10
8. 6

9. 12
10. 8
11. 3√13
12. 20√2
13. 6
14. 10√3
15. 2√5
16. 2√3
17. √6 / 3
18. 2√5 / 5

19. Rational
20. Irrational
21. Irrational
22. Irrational
23. Irrational
24. Irrational

25. ±4
26. ±7

27. If d = √(h² + k²), k missing → cannot compute.
But if d = √(h²) → d = 11.8 → ≈ 12 miles
28. Similarly → ≈ 216 miles

29. Radius ≈ 1080 miles

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⚠️ Note: Problems 27 and 28 are ambiguous due to missing information (k). Please confirm the full formula or values.

If you can provide the exact formula or clarify what k represents, I can refine those answers.

Let me know!