How many exercises are there in NCERT Class 8 Maths Chapter 6?

Chapter 6 (Cubes and Cube Roots) has 2 exercises. Exercise 6.1 covers properties of perfect cubes, interesting patterns like the sum of cubes identity, Hardy-Ramanujan numbers, and finding the smallest multiplier or divisor to make a number a perfect cube. Exercise 6.2 focuses on finding cube roots using prime factorization and the estimation method.

Can negative numbers be perfect cubes?

Yes. Unlike perfect squares, perfect cubes can be negative. For example, (-5) cubed = -125, so -125 is a perfect cube and the cube root of -125 is -5. This works because an odd number of negative factors gives a negative product.

What is a Hardy-Ramanujan number?

A Hardy-Ramanujan number can be expressed as the sum of two cubes in two different ways. The most famous example is 1729 = 1 cubed + 12 cubed = 9 cubed + 10 cubed. It is named after the famous anecdote involving mathematicians G.H. Hardy and S. Ramanujan about the cab number 1729.

How is this chapter different from Chapter 5?

Chapter 5 deals with squares (power 2) and square roots, while Chapter 6 deals with cubes (power 3) and cube roots. The prime factorization method is similar but groups factors in threes instead of twos. A key mathematical difference is that cube roots exist for negative numbers, but square roots of negative numbers are not real.

Is there a long division method for cube roots?

The NCERT textbook does not cover a long division method for cube roots at the Class 8 level. Prime factorization is the primary method taught for perfect cubes, and the estimation method is used for quickly finding cube roots of large perfect cubes of two-digit numbers.

How can SparkEd help me prepare for this chapter?

SparkEd offers adaptive practice at sparkedmaths.com/programs/8/cbse/squares-cubes with problems on perfect cubes, prime factorization, and estimation sorted by difficulty. The AI Math Solver provides step-by-step solutions. The AI Coach identifies weak areas and recommends targeted practice.

How many exercises are there in NCERT Class 8 Maths Chapter 6?

Chapter 6 (Cubes and Cube Roots) has 2 exercises. Exercise 6.1 covers properties of perfect cubes, interesting patterns like the sum of cubes identity, Hardy-Ramanujan numbers, and finding the smallest multiplier or divisor to make a number a perfect cube. Exercise 6.2 focuses on finding cube roots using prime factorization and the estimation method.

Can negative numbers be perfect cubes?

Yes. Unlike perfect squares, perfect cubes can be negative. For example, (-5) cubed = -125, so -125 is a perfect cube and the cube root of -125 is -5. This works because an odd number of negative factors gives a negative product.

What is a Hardy-Ramanujan number?

A Hardy-Ramanujan number can be expressed as the sum of two cubes in two different ways. The most famous example is 1729 = 1 cubed + 12 cubed = 9 cubed + 10 cubed. It is named after the famous anecdote involving mathematicians G.H. Hardy and S. Ramanujan about the cab number 1729.

How is this chapter different from Chapter 5?

Chapter 5 deals with squares (power 2) and square roots, while Chapter 6 deals with cubes (power 3) and cube roots. The prime factorization method is similar but groups factors in threes instead of twos. A key mathematical difference is that cube roots exist for negative numbers, but square roots of negative numbers are not real.

Is there a long division method for cube roots?

The NCERT textbook does not cover a long division method for cube roots at the Class 8 level. Prime factorization is the primary method taught for perfect cubes, and the estimation method is used for quickly finding cube roots of large perfect cubes of two-digit numbers.

How can SparkEd help me prepare for this chapter?

SparkEd offers adaptive practice at sparkedmaths.com/programs/8/cbse/squares-cubes with problems on perfect cubes, prime factorization, and estimation sorted by difficulty. The AI Math Solver provides step-by-step solutions. The AI Coach identifies weak areas and recommends targeted practice.

Solved Examples

NCERT Solutions for Class 8 Maths Chapter 6: Cubes and Cube Roots — Complete Guide

Complete solutions for all exercises — perfect cubes, interesting patterns, prime factorization, estimation, and 30+ solved problems.

CBSEClass 8

The SparkEd Authors (IITian & Googler)15 March 202650 min read

Why Cubes and Cube Roots Builds on Chapter 5

$Chapter 6 is the natural companion to Chapter 5 (Squares and Square Roots). While Chapter 5 dealt with the second power and its inverse, this chapter tackles the third power — cubes — and its inverse — cube roots. The methods and thinking patterns are remarkably similar, so if you have mastered Chapter 5, you are already halfway through Chapter 6.$

$n$

$What makes this chapter unique is the inclusion of some fascinating mathematical stories, especially Hardy-Ramanujan numbers (numbers that can be expressed as the sum of two cubes in two different ways). These add a layer of mathematical beauty to the otherwise computational chapter.$

$The chapter typically carries 4-6 marks in CBSE exams and connects directly to Chapter 5 (Squares), Chapter 10 (Exponents), and volume calculations in Chapter 9 (Mensuration). Let us dive into every concept and exercise!$

Perfect Cubes: Definition and Properties

$m$

$n$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$	$11$	$12$	$13$	$14$	$15$
$n^3$	$1$	$8$	$27$	$64$	$125$	$216$	$343$	$512$	$729$	$1000$	$1331$	$1728$	$2197$	$2744$	$3375$

$Key Properties:$

$4^3 = 64$

$(-3)^3 = -27$

$3$

$2 \leftrightarrow 8$

Solved Example 1: Checking Perfect Cubes

$392$

$Solution:$

392 = 2 \times 196 = 2 \times 2 \times 98 = 2^3 \times 49 = 2^3 \times 7^2

$7$

392 \times 7 = 2744 = 2^3 \times 7^3

\sqrt[3]{2744} = 2 \times 7 = 14

$7$

Interesting Patterns with Cubes

$Just like squares, cubes follow some beautiful patterns that are worth knowing for both exams and mathematical appreciation.$

Pattern 1: Sum of Consecutive Odd Numbers

$Every perfect cube can be expressed as a sum of consecutive odd numbers:$

1^3 = 1

2^3 = 3 + 5 = 8

3^3 = 7 + 9 + 11 = 27

4^3 = 13 + 15 + 17 + 19 = 64

5^3 = 21 + 23 + 25 + 27 + 29 = 125

$n^3$

Pattern 2: Sum of Cubes Formula

$n$

1^3 + 2^3 + 3^3 + \cdots + n^3 = \left(\dfrac{n(n+1)}{2}\right)^2

$1^3 + 2^3 = 1 + 8 = 9 = (1+2)^2 = 3^2$

$This elegant result is one of the most frequently tested patterns in CBSE exams.$

Pattern 3: Hardy-Ramanujan Numbers

$1729$

1729 = 1^3 + 12^3 = 1 + 1728

1729 = 9^3 + 10^3 = 729 + 1000

$1729$

Practice this topic on SparkEd — free visual solutions and AI coaching

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Cubes of Negative Numbers and Fractions

$Unlike squares (which are always non-negative), the cube of a negative number is negative :$

(-n)^3 = (-n) \times (-n) \times (-n) = -(n^3)

$(-2)^3 = -8$

$\sqrt[3]{-8} = -2$

$\sqrt{-4}$

$Cubes of Fractions:$

\left(\dfrac{a}{b}\right)^3 = \dfrac{a^3}{b^3}

$\left(\dfrac{2}{3}\right)^3 = \dfrac{8}{27}$

Solved Example 2: Cube of Negative Number

$(-7)^3$

$Solution:$

(-7)^3 = (-7) \times (-7) \times (-7) = 49 \times (-7) = -343

$(-7)^3 = -343$

Solved Example 3: Cube Root of Negative Number

$\sqrt[3]{-3375}$

$\sqrt[3]{-3375} = -\sqrt[3]{3375}$

$(-15)^3 = -3375$

Finding Cube Roots: Prime Factorization Method

$3$

$Steps: 1. Express the number as a product of prime factors. 2. Group identical factors in triples. 3. Take one factor from each triple and multiply them.$

$The result is the cube root.$

Solved Example 4: Cube Root by Prime Factorization

$\sqrt[3]{13824}$

$Solution:$

13824 = 2^9 \times 3^3

$(2^3)(2^3)(2^3)(3^3)$

\sqrt[3]{13824} = 2^3 \times 3 = 8 \times 3 = 24

$24^3 = 13824$

Solved Example 5: Smallest Multiplier for Perfect Cube

$1296$

$Solution:$

1296 = 2^4 \times 3^4

$3$

$2^2 \times 3^2 = 4 \times 9 = 36$

$1296 \times 36 = 46656 = 2^6 \times 3^6 = 36^3$

$36$

Solved Example 6: Smallest Divisor for Perfect Cube

$8788$

$Solution:$

8788 = 2^2 \times 13^3

$2$

\dfrac{8788}{4} = 2197 = 13^3

$\sqrt[3]{2197} = 13$

$4$

Solved Example 7: Cube Root of a Fraction

$\sqrt[3]{\dfrac{-216}{729}}$

$Solution:$

\sqrt[3]{\dfrac{-216}{729}} = \dfrac{\sqrt[3]{-216}}{\sqrt[3]{729}} = \dfrac{-6}{9} = \dfrac{-2}{3}

$(-6)^3 = -216$

$\dfrac{-2}{3}$

Finding Cube Roots: Estimation Method

$For large perfect cubes, you can estimate the cube root without full prime factorization. This method works for cubes of two-digit numbers.$

$2 \leftrightarrow 8$

Solved Example 8: Estimating Cube Root

$\sqrt[3]{175616}$

$Solution:$

$= 616$

$616$

$175$

$\sqrt[3]{175616} = 56$

$56^3 = 175616$

Solved Example 9: Another Estimation

$\sqrt[3]{474552}$

$Solution:$

$= 552$

$552$

$474$

$\sqrt[3]{474552} = 78$

$78^3 = 474552$

Exercise 6.1 — Complete Solutions

$Exercise 6.1 covers properties of perfect cubes, interesting patterns, and finding smallest multipliers/divisors.$

Solved Example 10: Is 500 a Perfect Cube?

$500$

$500 = 2^2 \times 5^3$

$2$

Solved Example 11: Sum of Cubes Identity

$1^3 + 2^3 + 3^3 + 4^3 + 5^3$

$1^3 + 2^3 + \cdots + n^3 = \left(\dfrac{n(n+1)}{2}\right)^2$

$= \left(\dfrac{5 \times 6}{2}\right)^2 = 15^2 = 225$

$1 + 8 + 27 + 64 + 125 = 225$

Solved Example 12: Express as Sum of Odd Numbers

$125$

$125 = 5^3 = 21 + 23 + 25 + 27 + 29$

$21 + 23 + 25 + 27 + 29 = 125$

Solved Example 13: Hardy-Ramanujan Verification

$4104$

$4104 = 2^3 + 16^3 = 8 + 4096 = 4104$

$4104$

Solved Example 14: Smallest Divisor

$3456$

$3456 = 2^7 \times 3^3$

$2$

$3456 \div 2 = 1728 = 12^3$

$2$

Exercise 6.2 — Complete Solutions

$Exercise 6.2 focuses on finding cube roots using prime factorization and estimation.$

Solved Example 15: Cube Root of Large Number

$\sqrt[3]{10648}$

$10648 = 2^3 \times 1331 = 2^3 \times 11^3$

$22^3 = 10648$

Solved Example 16: Cube Root of Product

$\sqrt[3]{27 \times 64}$

$\sqrt[3]{27 \times 64} = \sqrt[3]{27} \times \sqrt[3]{64} = 3 \times 4 = 12$

$12$

Solved Example 17: Decimal Cube Root

$\sqrt[3]{0.001331}$

$0.001331 = \dfrac{1331}{1000000} = \dfrac{11^3}{100^3}$

$\sqrt[3]{0.001331} = \dfrac{11}{100} = 0.11$

Solved Example 18: Word Problem — Volume

$5832$

$= \sqrt[3]{5832}$

$18^3 = 5832$

Solved Example 19: Estimation — Large Cube

$\sqrt[3]{32768}$

$768$

$\sqrt[3]{32768} = 32$

$32^3 = 32768$

Solved Example 20: Comparing Cube and Square

$1$

$n^6$

$1^6 = 1$

$1, 64,$

$64 = 8^2 = 4^3$

Common Mistakes Students Make

$1. Confusing square root and cube root groupings: * Mistake: Pairing factors in twos when finding cube root. * Fix: For cube roots, group in threes . For square roots, group in twos .$

$(-4)^3 = 64$

$2$

$4. Incorrect prime factorization: * Mistake: Missing a factor, leading to wrong cube root. * Fix: Always verify by cubing your answer.$

$5. Multiply vs. divide confusion: * Mistake: Multiplying when the problem asks to divide. * Fix: Read the problem carefully every time.$

$6. Not checking if all prime powers are multiples of 3: * Mistake: Declaring a number a perfect cube after partial factorization. * Fix: Complete the full prime factorization and check every prime's exponent.$

$7. Estimation method applied to non-two-digit roots: * Mistake: Using the estimation method for cube roots of three-digit numbers. * Fix: The estimation method as taught in Class 8 only works for cubes of two-digit numbers.$

Exam Strategy: How to Score Full Marks in Chapter 6

$Weightage: This chapter typically carries 4-6 marks in CBSE exams.$

$Typical Question Patterns:$

$729$

$* 2-3 Marks (SA): Find cube root by prime factorization; find smallest multiplier/divisor for a perfect cube; estimation of cube root; sum of cubes identity.$

$* 3-4 Marks (LA): Hardy-Ramanujan number problems; multi-step problems; expressing cubes as sums of odd numbers.$

$1^3 + 2^3 + \cdots + n^3 = \left(\dfrac{n(n+1)}{2}\right)^2$

$1$

$Practice on SparkEd's Squares & Cubes page!$

Quick Reference: Cubes from 1 to 15

$Memorise these — they appear in almost every exam:$

$n$	$n^3$	$n$	$n^3$
$1$	$1$	$9$	$729$
$2$	$8$	$10$	$1000$
$3$	$27$	$11$	$1331$
$4$	$64$	$12$	$1728$
$5$	$125$	$13$	$2197$
$6$	$216$	$14$	$2744$
$7$	$343$	$15$	$3375$
$8$	$512$

Connections to Other Chapters and Higher Classes

$^3$

$a^3 + b^3 = (a+b)(a^2 - ab + b^2)$

$In Class 10: Cube roots appear in volume problems involving combinations of solids and in some algebraic computations.$

$Mastering cubes now gives you a strong foundation for all these future topics!$

Boost Your Preparation with SparkEd

$You have covered the entire Cubes and Cube Roots chapter. Now put your knowledge into practice!$

$* Adaptive Practice: On our Squares & Cubes page, work through problems sorted by difficulty. * AI Math Solver: Stuck on a prime factorization or estimation problem? Get step-by-step solutions at sparkedmaths.com/solver . * AI Coach: Get personalized study recommendations based on your performance.$

$Head over to sparkedmaths.com and start practicing today!$

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NCERT Solutions for Class 8 Maths Chapter 6: Cubes and Cube Roots — Complete Guide

Why Cubes and Cube Roots Builds on Chapter 5

Perfect Cubes: Definition and Properties

Solved Example 1: Checking Perfect Cubes

Interesting Patterns with Cubes

Pattern 1: Sum of Consecutive Odd Numbers

Pattern 2: Sum of Cubes Formula

Pattern 3: Hardy-Ramanujan Numbers

Cubes of Negative Numbers and Fractions

Solved Example 2: Cube of Negative Number

Solved Example 3: Cube Root of Negative Number

Finding Cube Roots: Prime Factorization Method

Solved Example 4: Cube Root by Prime Factorization

Solved Example 5: Smallest Multiplier for Perfect Cube

Solved Example 6: Smallest Divisor for Perfect Cube

Solved Example 7: Cube Root of a Fraction

Finding Cube Roots: Estimation Method

Solved Example 8: Estimating Cube Root

Solved Example 9: Another Estimation

Exercise 6.1 — Complete Solutions

Solved Example 10: Is 500 a Perfect Cube?

Solved Example 11: Sum of Cubes Identity

Solved Example 12: Express as Sum of Odd Numbers

Solved Example 13: Hardy-Ramanujan Verification

Solved Example 14: Smallest Divisor

Exercise 6.2 — Complete Solutions

Solved Example 15: Cube Root of Large Number

Solved Example 16: Cube Root of Product

Solved Example 17: Decimal Cube Root

Solved Example 18: Word Problem — Volume

Solved Example 19: Estimation — Large Cube

Solved Example 20: Comparing Cube and Square

Common Mistakes Students Make

Exam Strategy: How to Score Full Marks in Chapter 6

Quick Reference: Cubes from 1 to 15

Connections to Other Chapters and Higher Classes

Boost Your Preparation with SparkEd

Practice These Topics on SparkEd

Frequently Asked Questions

How many exercises are there in NCERT Class 8 Maths Chapter 6?

Can negative numbers be perfect cubes?

What is a Hardy-Ramanujan number?

How is this chapter different from Chapter 5?

Is there a long division method for cube roots?

How can SparkEd help me prepare for this chapter?

Try SparkEd Free

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