CUBES AND CUBE-ROOTS

Subject: Mathematics — Topic: Numbers — Target age: 14 (Kenyan secondary)

Learning objectives

• Understand what a cube of a number is and how to calculate it.
• Know perfect cubes and use them to estimate cube-roots.
• Find cube-roots of perfect cubes by inspection and prime-factor method.
• Work with cubes and cube-roots of whole numbers, negatives and fractions.

1. Definition

The cube of a number a is a×a×a and is written as a³. The cube-root of a number x, written ∛x, is the number that when cubed gives x. So ∛(a³) = a and (∛x)³ = x.

2. Examples of cubes

Some perfect cubes (useful to memorise):

3. A simple visual: 2³ = 8 (a 2×2×2 cube)

There are 2 layers like this, so total unit cubes = 2×2×2 = 8.

4. Cube of negative numbers

Cubing keeps the sign: (-a)³ = -(a³). Example: (-3)³ = -27 so ∛(-27) = -3.

5. How to find cube-root of a perfect cube

1. By inspection: recall perfect cubes (e.g., ∛125 = 5).
2. By prime factorisation:

   Example: ∛216. Factor 216 = 2³ × 3³. Group factors in threes: (2×3) = 6 → ∛216 = 6.

3. Using exponents: ∛x = x^(1/3). So if x = a³ then x^(1/3) = a.

6. Cube-root of non-perfect cubes (estimation)

Find nearest perfect cubes and interpolate. Example: ∛50.

3³ = 27 and 4³ = 64 so ∛50 is between 3 and 4. Since 50 is closer to 64, ∛50 ≈ 3.7 (calculator: ≈3.684).

7. Cube-root of fractions

For positive fractions: ∛(a/b) = ∛a / ∛b. Example: ∛(8/27) = 2/3.

8. Properties to remember

• (ab)³ = a³ b³
• ∛(ab) = ∛a × ∛b (for all real a, b when cube-roots are real)
• ∛(a³) = a for every real a (odd root, so works for negatives)
• Cubes of last digits (0–9) give patterns useful for quick checks: 0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9 (these are the last digits of 0³..9³).

9. Worked examples

10. Practice questions

1. Find the cube of 9.
2. Find ∛216 and ∛512.
3. Find ∛(-27).
4. Find ∛(64/125).
5. Use prime factors to find ∛(27000).
6. Estimate ∛300 (give one decimal place).
7. Is 345 a perfect cube? Explain briefly.

Answers (click to show)

1. 9³ = 729.
2. ∛216 = 6, ∛512 = 8.
3. ∛(-27) = -3.
4. ∛(64/125) = 4/5 = 0.8.
5. 27000 = 27 × 1000 = 3³ × 10³ = (3×10)³ = 30³ so ∛27000 = 30.
6. 5³ = 125, 6³ = 216, 7³ = 343 so ∛300 between 6 and 7, closer to 7. Approx ≈ 6.7.
7. 345 is not a perfect cube because it is not equal to n³ for any integer n. Nearest cubes are 7³ = 343 and 8³ = 512, so 345 is not a perfect cube.

Study tips

• Memorise cubes up to at least 12³ = 1728.
• Practice prime factorisation to extract cube roots quickly.
• Use estimation (between two perfect cubes) for non-perfect cubes.

Prepared for Kenyan learners (age 14). If you want more practice problems or step-by-step prime factorisation examples, say which question to expand.