Cube Roots from 1-50: Complete Guide with Simple Examples for Students

What is a Cube Root?

A cube root of a number is the value that, when multiplied by itself three times, gives the original number. We write the cube root using the symbol ? (or as a fractional exponent of ?).

Mathematically, if ?x = y, then y³ = x.

Think of it this way: if cubing is the process of "building a cube," then finding the cube root is "finding the length of one side of that cube."

Perfect Cube Roots

Some numbers are "perfect cubes," meaning they are the result of cubing a whole number. Finding their cube roots gives exact answers:

Cube Root of 1 (?1)

?1 = 1 (because 1³ = 1)

• Example: A cube with volume 1 cubic inch has sides of 1 inch.

Cube Root of 8 (?8)

?8 = 2 (because 2³ = 8)

• Example: A cube with volume 8 cubic centimeters has sides of 2 centimeters.

Cube Root of 27 (?27)

?27 = 3 (because 3³ = 27)

• Example: A cube-shaped box with volume 27 cubic feet has sides of 3 feet.

Cube Root of 64 (?64)

?64 = 4 (because 4³ = 64)

• Example: A cube with volume 64 cubic meters has sides measuring 4 meters each.

Cube Root of 125 (?125)

?125 = 5 (because 5³ = 125)

• Example: A cube-shaped tank holding 125 gallons has sides of 5 units (in the appropriate measurement).

Cube Roots from 1 to 50

Let's explore the cube roots of numbers from 1 to 50, highlighting perfect cubes and approximating others:

1. Cube Root of 1 (?1) = 1

Perfect cube: 1³ = 1

2. Cube Root of 2 (?2) ≈ 1.2599

Not a perfect cube.

• Example: If you had a cube with volume 2 cubic inches, each side would be approximately 1.26 inches.

3. Cube Root of 3 (?3) ≈ 1.4422

Not a perfect cube.

• Example: A cube-shaped container with volume 3 cubic feet would have sides of about 1.44 feet.

4. Cube Root of 4 (?4) ≈ 1.5874

Not a perfect cube.

• Example: A cube with volume 4 cubic meters has sides of approximately 1.59 meters.

5. Cube Root of 5 (?5) ≈ 1.7100

Not a perfect cube.

• Example: A cube-shaped package with volume 5 cubic units has sides of about 1.71 units.

6. Cube Root of 6 (?6) ≈ 1.8171

Not a perfect cube.

• Example: A cube with volume 6 cubic centimeters has sides measuring about 1.82 centimeters.

7. Cube Root of 7 (?7) ≈ 1.9129

Not a perfect cube.

• Example: A cube-shaped block with volume 7 cubic inches has sides of approximately 1.91 inches.

8. Cube Root of 8 (?8) = 2

Perfect cube: 2³ = 8

• Example: A sugar cube with volume 8 cubic centimeters has sides exactly 2 centimeters long.

9. Cube Root of 9 (?9) ≈ 2.0801

Not a perfect cube.

• Example: A cube with volume 9 cubic units has sides of about 2.08 units.

10. Cube Root of 10 (?10) ≈ 2.1544

Not a perfect cube.

• Example: A cube-shaped container with volume 10 cubic feet has sides of approximately 2.15 feet.

11. Cube Root of 11 (?11) ≈ 2.2240

Not a perfect cube.

• Example: A cube with volume 11 cubic meters has sides of about 2.22 meters.

12. Cube Root of 12 (?12) ≈ 2.2894

Not a perfect cube.

• Example: A cube-shaped box with volume 12 cubic inches has sides measuring about 2.29 inches.

13. Cube Root of 13 (?13) ≈ 2.3513

Not a perfect cube.

• Example: A cube with volume 13 cubic centimeters has sides of approximately 2.35 centimeters.

14. Cube Root of 14 (?14) ≈ 2.4101

Not a perfect cube.

• Example: A cube-shaped container with volume 14 cubic units has sides of about 2.41 units.

15. Cube Root of 15 (?15) ≈ 2.4662

Not a perfect cube.

• Example: A cube with volume 15 cubic feet has sides of approximately 2.47 feet.

16. Cube Root of 16 (?16) ≈ 2.5198

Not a perfect cube.

• Example: A cube-shaped package with volume 16 cubic meters has sides of about 2.52 meters.

17. Cube Root of 17 (?17) ≈ 2.5713

Not a perfect cube.

• Example: A cube with volume 17 cubic inches has sides measuring about 2.57 inches.

18. Cube Root of 18 (?18) ≈ 2.6207

Not a perfect cube.

• Example: A cube-shaped box with volume 18 cubic centimeters has sides of approximately 2.62 centimeters.

19. Cube Root of 19 (?19) ≈ 2.6684

Not a perfect cube.

• Example: A cube with volume 19 cubic units has sides of about 2.67 units.

20. Cube Root of 20 (?20) ≈ 2.7144

Not a perfect cube.

• Example: A cube-shaped container with volume 20 cubic feet has sides of approximately 2.71 feet.

21. Cube Root of 21 (?21) ≈ 2.7589

Not a perfect cube.

• Example: A cube with volume 21 cubic meters has sides of about 2.76 meters.

22. Cube Root of 22 (?22) ≈ 2.8020

Not a perfect cube.

• Example: A cube-shaped box with volume 22 cubic inches has sides measuring about 2.80 inches.

23. Cube Root of 23 (?23) ≈ 2.8439

Not a perfect cube.

• Example: A cube with volume 23 cubic centimeters has sides of approximately 2.84 centimeters.

24. Cube Root of 24 (?24) ≈ 2.8845

Not a perfect cube.

• Example: A cube-shaped container with volume 24 cubic units has sides of about 2.88 units.

25. Cube Root of 25 (?25) ≈ 2.9240

Not a perfect cube.

• Example: A cube with volume 25 cubic feet has sides measuring about 2.92 feet.

26. Cube Root of 26 (?26) ≈ 2.9625

Not a perfect cube.

• Example: A cube-shaped box with volume 26 cubic meters has sides of approximately 2.96 meters.

27. Cube Root of 27 (?27) = 3

Perfect cube: 3³ = 27

• Example: A Rubik's cube has a volume of 27 smaller cubes and sides of length 3 cubes.

28. Cube Root of 28 (?28) ≈ 3.0365

Not a perfect cube.

• Example: A cube with volume 28 cubic inches has sides of about 3.04 inches.

29. Cube Root of 29 (?29) ≈ 3.0723

Not a perfect cube.

• Example: A cube-shaped container with volume 29 cubic centimeters has sides measuring about 3.07 centimeters.

30. Cube Root of 30 (?30) ≈ 3.1072

Not a perfect cube.

• Example: A cube with volume 30 cubic units has sides of approximately 3.11 units.

31. Cube Root of 31 (?31) ≈ 3.1413

Not a perfect cube.

• Example: A cube-shaped box with volume 31 cubic feet has sides of about 3.14 feet.

32. Cube Root of 32 (?32) ≈ 3.1748

Not a perfect cube.

• Example: A cube with volume 32 cubic meters has sides measuring about 3.17 meters.

33. Cube Root of 33 (?33) ≈ 3.2075

Not a perfect cube.

• Example: A cube-shaped container with volume 33 cubic inches has sides of approximately 3.21 inches.

34. Cube Root of 34 (?34) ≈ 3.2396

Not a perfect cube.

• Example: A cube with volume 34 cubic centimeters has sides of about 3.24 centimeters.

35. Cube Root of 35 (?35) ≈ 3.2711

Not a perfect cube.

• Example: A cube-shaped box with volume 35 cubic units has sides measuring about 3.27 units.

36. Cube Root of 36 (?36) ≈ 3.3019

Not a perfect cube.

• Example: A cube with volume 36 cubic feet has sides of approximately 3.30 feet.

37. Cube Root of 37 (?37) ≈ 3.3322

Not a perfect cube.

• Example: A cube-shaped container with volume 37 cubic meters has sides of about 3.33 meters.

38. Cube Root of 38 (?38) ≈ 3.3620

Not a perfect cube.

• Example: A cube with volume 38 cubic inches has sides measuring about 3.36 inches.

39. Cube Root of 39 (?39) ≈ 3.3912

Not a perfect cube.

• Example: A cube-shaped box with volume 39 cubic centimeters has sides of approximately 3.39 centimeters.

40. Cube Root of 40 (?40) ≈ 3.4200

Not a perfect cube.

• Example: A cube with volume 40 cubic units has sides of about 3.42 units.

41. Cube Root of 41 (?41) ≈ 3.4482

Not a perfect cube.

• Example: A cube-shaped container with volume 41 cubic feet has sides measuring about 3.45 feet.

42. Cube Root of 42 (?42) ≈ 3.4760

Not a perfect cube.

• Example: A cube with volume 42 cubic meters has sides of approximately 3.48 meters.

43. Cube Root of 43 (?43) ≈ 3.5034

Not a perfect cube.

• Example: A cube-shaped box with volume 43 cubic inches has sides of about 3.50 inches.

44. Cube Root of 44 (?44) ≈ 3.5303

Not a perfect cube.

• Example: A cube with volume 44 cubic centimeters has sides measuring about 3.53 centimeters.

45. Cube Root of 45 (?45) ≈ 3.5569

Not a perfect cube.

• Example: A cube-shaped container with volume 45 cubic units has sides of approximately 3.56 units.

46. Cube Root of 46 (?46) ≈ 3.5830

Not a perfect cube.

• Example: A cube with volume 46 cubic feet has sides of about 3.58 feet.

47. Cube Root of 47 (?47) ≈ 3.6088

Not a perfect cube.

• Example: A cube-shaped box with volume 47 cubic meters has sides measuring about 3.61 meters.

48. Cube Root of 48 (?48) ≈ 3.6342

Not a perfect cube.

• Example: A cube with volume 48 cubic inches has sides of approximately 3.63 inches.

49. Cube Root of 49 (?49) ≈ 3.6593

Not a perfect cube.

• Example: A cube-shaped container with volume 49 cubic centimeters has sides of about 3.66 centimeters.

50. Cube Root of 50 (?50) ≈ 3.6840

Not a perfect cube.

• Example: A cube with volume 50 cubic units has sides measuring about 3.68 units.

How to Calculate Cube Roots

There are several methods to find cube roots:

1. Using a Calculator

Most scientific calculators have a cube root function (often labeled as "?" or "³√").

2. Estimation Method

For numbers between perfect cubes, you can estimate:

• Example: To find ?30, note that 27 < 30 < 64, so 3 < ?30 < 4.
• Since 30 is closer to 27 than 64, ?30 is closer to 3, approximately 3.11.

3. Prime Factorization Method

Break the number into prime factors and group them in threes:

• Example: Find ?216
  • 216 = 2³ × 3³ = (2 × 3)³ = 6³
  • Therefore, ?216 = 6

4. Perfect Cube Recognition

Memorize perfect cubes to quickly identify their cube roots:

• 1³ = 1, so ?1 = 1
• 2³ = 8, so ?8 = 2
• 3³ = 27, so ?27 = 3
• 4³ = 64, so ?64 = 4
• 5³ = 125, so ?125 = 5

Perfect Cubes from 1 to 50

Within our range of 1 to 50, only four numbers are perfect cubes:

1. 1 (1³)
2. 8 (2³)
3. 27 (3³)
4. Cube Root of 64 (4³) is not in our range, as it equals 64 which is > 50

Real-Life Applications of Cube Roots

1. Engineering: Calculating dimensions of cube-shaped structures from their volumes
2. Physics: Determining the relationship between volume and linear dimensions
3. Architecture: Designing proportional spaces based on volume requirements
4. Computer graphics: Scaling 3D models proportionally in three dimensions
5. Material science: Relating density and volume to linear measurements

Patterns in Cube Roots

1. The cube root of a negative number is negative:
   • ?(-8) = -2, because (-2)³ = -8
2. The cube root function is continuous and strictly increasing:
   • If a < b, then ?a < ?b
3. The difference between consecutive cube roots decreases as numbers increase:
   • ?2 - ?1 ≈ 0.26
   • ?3 - ?2 ≈ 0.18
   • ?4 - ?3 ≈ 0.15

Practice Problems

1. Find the cube root of 27.
2. Estimate the cube root of 15 without a calculator.
3. If the volume of a cube is 42 cubic meters, what is the approximate length of one side?
4. Which is greater: ?20 or ?22?
5. Find the cube root of 125.

Conclusion

Understanding cube roots helps us solve various mathematical and real-world problems involving three-dimensional objects. Remember that the cube root of a number represents the side length of a cube with that given volume. While perfect cubes have exact cube roots, we can estimate or calculate approximate cube roots for other numbers using various methods.

When working with cube roots, practice recognizing perfect cubes and using estimation techniques to develop your mathematical intuition. The cube root function appears throughout mathematics, science, and engineering, making it an essential concept to master