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Class 8 Mathematics Chapter 6 Cubes and Cube Roots — Formulas & Key Points

Chapter 6 Cubes and Cube Roots extends the idea of squares and square roots to the third power. Understanding cubes is crucial for algebra, volume calculations and higher mathematics. This formula sheet brings together every definition, identity, method and shortcut from the NCERT Class 8 Mathematics syllabus into one ready-reference page, saving hours during revision and ensuring no formula is forgotten on exam day.

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Key takeaways

✓A perfect cube is formed when a number is multiplied by itself three times; symbol n³ means n × n × n.
✓Cube root by prime factorisation groups factors into triplets; each triplet contributes one factor to the cube root.
✓The cube of an even number is always even; the cube of an odd number is always odd.
✓Adding consecutive odd numbers starting from 1 gives perfect cubes: 1=1³, 1+7=8=2³, 1+7+19=27=3³.
✓Units digit of a cube depends only on the units digit of the base number following a fixed pattern.
✓Cube roots of negative numbers are negative; ∛(-n) = -∛n.
✓Column method for cube root works for larger numbers by grouping digits in threes from the right.

Core Definitions and Notation

Before working with formulas, clarity on terminology is essential. A cube is the result of multiplying a number by itself twice more. The cube of n is written as n³ (read as 'n cubed') and equals n × n × n. The inverse operation is the cube root, denoted by the symbol ∛ or by writing the number under a radical with index 3. If a³ = b, then ∛b = a. A perfect cube is any integer that can be expressed as the cube of another integer, such as 1, 8, 27, 64, 125, and so on. Non-perfect cubes like 10 or 50 have irrational cube roots that cannot be expressed as simple fractions. Understanding these definitions helps in recognising which method to apply during problem-solving.

Cube of n: n³ = n × n × n
Cube root of b: ∛b = a if and only if a³ = b
Perfect cube: an integer m such that m = k³ for some integer k
Symbol notation: ∛ is the cube root symbol; index 3 may be written explicitly

Table of Perfect Cubes (1 to 20)

Memorising the cubes of numbers from 1 to 20 saves calculation time and helps in instant pattern recognition during exams. This table is particularly useful when factorising larger numbers or estimating cube roots. Students should practice writing these from memory at least once a week. Knowing that 10³ = 1000 and 20³ = 8000 helps bracket cube roots of four-digit numbers quickly. The CBSE Class 8 Mathematics paper often includes MCQs where recognising a perfect cube instantly can save precious seconds. Many students find it helpful to write this table on the first page of their answer sheet as soon as the exam starts, ensuring they have a reference throughout the paper without needing to recalculate.

1³=1, 2³=8, 3³=27, 4³=64, 5³=125, 6³=216, 7³=343, 8³=512, 9³=729, 10³=1000
11³=1331, 12³=1728, 13³=2197, 14³=2744, 15³=3375
16³=4096, 17³=4913, 18³=5832, 19³=6859, 20³=8000

Properties and Identities of Cubes

Several algebraic identities and properties govern cubes. The most frequently used in CBSE exams are the sum and difference of cubes formulas, which factorise expressions efficiently. The identity (a + b)³ = a³ + 3a²b + 3ab² + b³ expands a binomial cube, while a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²) factorise sums and differences. These are tested in algebraic manipulation questions and also appear in higher classes. Another key property is that the cube of a sum is not equal to the sum of cubes unless one term is zero: (a+b)³ ≠ a³+b³. Recognising when to expand and when to factorise is a skill built through practice with NCERT exercises and CBSE sample papers.

(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
(−n)³ = −n³ for any real number n

Cube Root by Prime Factorisation Method

This is the most reliable method taught in NCERT Class 8 Mathematics for finding cube roots of perfect cubes. The steps are systematic: first, resolve the number into prime factors using repeated division. Write the factorisation in exponent form. Then group the prime factors into triplets (sets of three identical factors). For each triplet, take one factor out. Multiply these factors to obtain the cube root. If any prime factor is left ungrouped (appearing once or twice), the original number is not a perfect cube and this method will not yield an integer cube root. This technique is tested in CBSE board exams through both short-answer and long-answer questions, often combined with word problems involving volumes of cubes.

Step 1: Factorise the number into primes
Step 2: Group identical primes into triplets
Step 3: Take one factor from each triplet
Step 4: Multiply these factors to get the cube root
If factors remain ungrouped, the number is not a perfect cube

Column Method (Estimation) for Cube Roots

The column method, also called the digit-by-digit or estimation method, is useful for finding cube roots of large perfect cubes without a calculator. The procedure begins by grouping the digits of the number into periods of three, starting from the units place and moving left. The first period (rightmost group) determines the units digit of the cube root using the units-digit pattern table. The second period (left group) is compared with the cube table to estimate the tens digit. This method is particularly helpful when prime factorisation becomes tedious for six-digit or larger numbers. While NCERT introduces this method for awareness, most CBSE exam questions are designed to be solved via prime factorisation or direct recognition from the memorised table of cubes up to 20.

Step 1: Group digits in threes from right to left
Step 2: Use the rightmost group to find units digit of cube root
Step 3: Use the leftmost group to estimate tens digit by comparing with cube table
Step 4: Combine the digits to write the cube root

Units Digit Pattern in Cubes

The units digit of a cube depends solely on the units digit of the base number and follows a predictable cycle. This pattern is a powerful shortcut for multiple-choice questions and quick checks. If a number ends in 0, its cube ends in 0. If it ends in 1, the cube ends in 1. A number ending in 2 gives a cube ending in 8, and one ending in 3 gives a cube ending in 7. The digit 4 cubes to end in 4, 5 to 5, 6 to 6, 7 to 3, 8 to 2, and 9 to 9. Notice the symmetry: 2↔8, 3↔7. Memorising this table allows instant verification of whether a proposed cube root is correct by checking only the last digit, saving time in CBSE board exams and competitive tests.

0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9

Pattern of Sums of Consecutive Odd Numbers

An elegant property linking cubes and odd numbers is that the cube of any natural number n equals the sum of n consecutive odd numbers. Specifically, n³ is the sum of n odd numbers starting from (n² − n + 1). For example, 1³=1, 2³=3+5=8, 3³=7+9+11=27, 4³=13+15+17+19=64. This pattern is derived from the difference of consecutive cubes: (n+1)³ − n³ = 3n² + 3n + 1, which is always odd. While not directly a formula to compute cubes, this property deepens conceptual understanding and occasionally appears in CBSE reasoning questions or Olympiad-level problems. Recognising it helps students appreciate the structure within number theory and builds algebraic intuition for higher classes where such identities recur in polynomial expansions.

1³ = 1
2³ = 3 + 5 = 8
3³ = 7 + 9 + 11 = 27
4³ = 13 + 15 + 17 + 19 = 64
n³ is the sum of n consecutive odd numbers beginning at n²−n+1

Common Mistakes and Notation Pitfalls

Students frequently confuse cubes with squares, writing n³ when they mean n² or vice versa, especially under exam pressure. Another common error is misapplying the cube root symbol: writing ∛(a+b) = ∛a + ∛b, which is mathematically incorrect. Cube roots do not distribute over addition or subtraction. When finding cube roots by prime factorisation, forgetting to group all three identical factors leads to wrong answers; each triplet must be complete. Sign errors are also frequent: the cube of a negative number is negative, so ∛(-27) = -3, not +3. In CBSE marking schemes, one mark is often deducted for incorrect notation or sign. Double-checking the sign, ensuring triplets are correctly grouped, and verifying the final answer by cubing it back are habits that prevent these pitfalls and secure full marks in Class 8 Mathematics exams.

Do NOT write ∛(a+b) = ∛a + ∛b; cube roots do not distribute
Cube of negative is negative: (-n)³ = -n³
Ensure every triplet is complete in prime factorisation; leftover factors mean not a perfect cube
Units digit shortcut applies only to the last digit, not the entire number
Always verify your cube root by cubing the answer

Memory Tricks and Mnemonics

Remembering the cubes from 1 to 10 can be made easier with patterns and rhymes. Notice that 1³=1, 10³=1000 (just add three zeros). For middle values, use the fact that (10−n)³ and (10+n)³ are symmetrically spaced around 1000. The units-digit cycle (0,1,8,7,4,5,6,3,2,9) can be memorised using the mnemonic 'Zero One Ate Seven, Four Five Six, Three Two Nine'. Some students create a mental image of a dice (6 faces) to recall that 6³=216 (six cubed is two-one-six). For the sum and difference identities, remember 'plus cubed has all plus signs in the expansion, minus cubed alternates signs'. Flashcards, daily oral drills, and timed quizzes on perfect cubes significantly improve recall speed, which is critical for CBSE Class 8 Mathematics term exams where every minute counts.

Cubes 1-10: 1,8,27,64,125,216,343,512,729,1000
Units pattern mnemonic: 'Zero One Ate(8) Seven, Four Five Six, Three Two Nine'
Identity tip: (a+b)³ all terms positive; (a−b)³ alternates signs
Dice visual: 6³=216 (six, two-one-six)

Solved Mini-Examples Applying the Formulas

Working through examples cements formula application. Example A: Is 392 a perfect cube? Factorise 392 = 2×2×2 × 7×7. Group: (2,2,2) is one triplet; (7,7) is incomplete. Hence 392 is not a perfect cube. Example B: Find the smallest number by which 2400 must be multiplied to get a perfect cube. Factorise 2400 = 2⁴×3×5². Triplets need three of each prime. 2 needs two more (to make 2⁶), 3 needs two more, 5 needs one more. Multiply by 2²×3²×5 = 4×9×5=180. So 2400×180=432000, a perfect cube. Example C: Simplify (∛512 + ∛(-125)). ∛512=8, ∛(-125)=-5, so sum is 8+(-5)=3. These examples mirror CBSE exam patterns and practicing them sharpens speed and accuracy during real tests.

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Last-Minute Revision Checklist

Before entering the exam hall, run through this quick checklist to ensure every formula is at your fingertips. Write the cubes from 1 to 20 on scrap paper. Recall the units-digit pattern (0→0, 1→1, 2→8, 3→7, 4→4, 5→5, 6→6, 7→3, 8→2, 9→9). Recite the sum and difference identities: a³+b³=(a+b)(a²−ab+b²) and a³−b³=(a−b)(a²+ab+b²). Remember the expansion (a+b)³=a³+3a²b+3ab²+b³. Check your understanding: can you factorise a six-digit perfect cube by prime factorisation in under three minutes? Can you apply the column method to estimate cube roots? Do you know that the cube of a negative number is negative? Review common mistakes: cube root does not distribute over addition. This five-minute drill boosts confidence and prevents silly errors worth two to three marks on the CBSE paper.

Write cubes 1–20 from memory
Recite units-digit pattern for cubes
State a³±b³ identities aloud
Verify one cube root by prime factorisation
Check sign rules: (−n)³ = −n³
Confirm: ∛(a+b) ≠ ∛a + ∛b

Frequently asked questions

What is the difference between a cube and a cube root?+

A cube is the result of multiplying a number by itself three times (n³=n×n×n), while a cube root is the inverse operation that finds which number, when cubed, gives the original value (∛n³=n). For example, 5³=125 and ∛125=5.

How do I quickly check if a large number is a perfect cube?+

Factorise the number into prime factors. If every prime factor appears in groups of three (exponent divisible by 3), it is a perfect cube. Any leftover factors (exponent not divisible by 3) mean it is not a perfect cube.

Why is the cube of a negative number negative?+

When you multiply a negative number by itself three times, you get (−n)×(−n)×(−n). The first two negatives give a positive, then multiplying by the third negative makes the result negative again. Hence (−2)³=−8.

What is the units-digit trick and how does it help in exams?+

The units digit of any cube depends only on the units digit of the base number, following the pattern 0→0,1→1,2→8,3→7,4→4,5→5,6→6,7→3,8→2,9→9. This lets you verify answers instantly without full calculation, saving time in CBSE exams.

Can I use a calculator to find cube roots in CBSE exams?+

No, CBSE Class 8 Mathematics exams do not allow calculators. You must use prime factorisation, the column method, or recognise perfect cubes from memory. Practicing these methods is essential for scoring full marks.

How many cubes should I memorise for the Class 8 exam?+

Memorise cubes from 1 to 20 at minimum. Knowing up to 20³=8000 covers most CBSE textbook and exam questions. Some toppers memorise up to 25 or 30 for extra confidence and speed in competitive tests.

What if the prime factorisation leaves factors that do not form triplets?+

If after grouping prime factors into triplets some factors remain (appear once or twice), the number is not a perfect cube. You cannot find an integer cube root. The question may ask you to find the smallest multiplier to make it a perfect cube.

Is the sum of two cubes equal to the cube of their sum?+

No. In general, a³+b³ ≠ (a+b)³. For example, 1³+2³=1+8=9, but (1+2)³=27. The identity for sum of cubes is a³+b³=(a+b)(a²−ab+b²), which is different from (a+b)³.

How does CBSETUTOR.ai help with Chapter 6 doubts at odd hours?+

CBSETUTOR.ai is available 24×7. If your child is revising late at night and gets stuck on a cube-root problem, they upload a photo and receive a step-by-step NCERT-aligned solution instantly. The AI tutor covers all chapters for Classes 6-12 at ₹999/month with a 3-day free trial.

What is the column method for cube roots and when should I use it?+

The column method groups the digits of a number into threes from the right, then uses the units-digit pattern and the cube table to estimate each digit of the cube root. Use it for large perfect cubes when prime factorisation is too lengthy, though CBSE usually designs questions solvable by factorisation.

Related resources

Important Questions: CBSE Class 8 Mathematics Chapter 6 Cubes and Cube Roots CBSE Class 8 Mathematics Chapter 6 Cubes and Cube Roots Worksheet with Answers CBSE Class 8 Mathematics Chapter 5 Squares and Square Roots Worksheet with Answers Class 8 Mathematics Chapter 7 Comparing Quantities — Formulas & Key Points AI Tutor for Class 8: The Smart Alternative to Tuition AI Tutor for Class 8 Social Science: Learn Faster with Instant Help NCERT Solutions for Class 9 Mathematics Chapter 1: Number Systems – Complete Guide Class 9 Mathematics Chapter 2 Polynomials — Formulas & Key Points

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