Class 8 Mathematics Cubes and Cube Roots 

 

These notes of Cubes and Cube Roots cover the entire NCERT Class 8 Chapter 7 (earlier Chapter 7, now Chapter 6 in some revised editions). Practise at least 20–30 cube root questions using the estimation method to master the chapter.

1. Introduction to Cubes: Cubes and Cube Roots

• When a number is multiplied by itself three times, the product is called the cube of that number.
• Cube of a number ‘a’ is written as a³.
• Formula: a³ = a × a × a Example: 2³ = 2 × 2 × 2 = 8 5³ = 5 × 5 × 5 = 125 10³ = 10 × 10 × 10 = 1000

2. Properties of Cubes

1. Cubes of even numbers are always even. Ex: 4³ = 64 (even), 6³ = 216 (even)
2. Cubes of odd numbers are always odd. Ex: 3³ = 27 (odd), 7³ = 343 (odd)
3. Cubes of negative numbers are negative. Ex: (-2)³ = -8, (-5)³ = -125
4. Cube of 0 is 0.
5. The cube of a number ending with 0 ends with 0. Ex: 10³ = 1000, 20³ = 8000

3. Interesting Patterns in Units Digit of Cubes

The units digit of a cube depends only on the units digit of the original number:

Very useful for quick mental calculation and finding cube roots.

4. Perfect Cubes

• A natural number is called a perfect cube if it is the cube of some natural number.
• Examples: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, … are perfect cubes.
• These are cubes of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …

5. Cube Roots

• The cube root of a number ‘x’ is a number which when multiplied by itself three times gives ‘x’.
• Symbol: ∛x or x^(1/3)
• Example: ∛8 = 2 (because 2 × 2 × 2 = 8) ∛125 = 5 ∛1000 = 10 ∛(-64) = -4 (cube root of negative number is negative)

6. Cube Root of a Perfect Cube

• Every perfect cube has an exact cube root which is an integer.
• Cube root of non-perfect cubes is generally not an integer (e.g., ∛2 ≈ 1.26).

7. Method 1: Finding Cube Root by Prime Factorisation

Steps:

1. Express the number as a product of prime factors.
2. Make groups of three identical factors.
3. Take one factor from each group and multiply them. That gives the cube root.

Example 1: Find ∛(3375) 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3³ × 5³ ∛3375 = 3 × 5 = 15

Example 2: ∛(21952) 21952 = 2 × 2 × 2 × 13 × 13 × 13 = 2³ × 13³ ∛21952 = 2 × 13 = 26

Example 3: ∛(1728) 1728 = 2³ × 2³ × 3³ = (2 × 2 × 3)³ = 12³ ∛1728 = 12

8. Method 2: Estimation Method (Short-cut for Cube Roots of Perfect Cubes)

This is the fastest method for large perfect cubes in exams.

Steps:

1. Look at the units digit of the number and match it with the units digit table (above) to find possible units digit of cube root.
2. Remove the last three digits (thousands place and beyond) and look at the remaining number.
3. Find between which two cubes it lies.
4. Choose the smaller or larger number accordingly.

Example: Find ∛(59319) Step 1: Units digit = 9 → cube root must end with 9 (only 9³ ends with 9) Step 2: Remove last three digits → 59 Step 3: Find cubes: 3³ = 27 4³ = 64 So 27 < 59 < 64 → cube root lies between 30 and 40 Step 4: Since we already know units digit is 9 → answer = 39 Verification: 39³ = 39 × 39 × 39 = 1521 × 39 = 59319 ✓

Another Example: ∛(91125)

• Units digit 5 → cube root ends with 5
• Remove last 3 digits → 91
• 4³ = 64, 5³ = 125 → 64 < 91 < 125 → tens digit = 4
• So cube root = 45 Check: 45³ = 45 × 45 × 45 = 2025 × 45 = 91125 ✓

9. Quick Reference Table of Cubes (1 to 20) – Must Memorise

10. Cube Roots of Numbers Less Than 1 (Decimals)

• ∛0.001 = 0.1 (because 0.1 × 0.1 × 0.1 = 0.001)
• ∛0.000027 = 0.03
• ∛0.216 = 0.6

11. Cube Root of Product and Quotient

• ∛(a × b) = ∛a × ∛b
• ∛(a / b) = ∛a / ∛b (b ≠ 0)

12. Important Points for Exams

• If a number has 4 or 5 digits → cube root has 2 digits.
• If a number has 6 or 7 digits → cube root has 3 digits.
• Always check if the number is a perfect cube by estimation method first.
• For non-perfect cubes, we usually say cube root is not an integer (unless asked for approximate value).

13. Summary Table for Quick Revision

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Chapter 8: Comparing Quantities