Class 8 Mathematics Chapter 6 Cubes and Cube Roots

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I. Chapter Summary:

This chapter introduces students to the concept of cubes and cube roots of natural numbers. It explores the properties and patterns of perfect cubes and their identification using prime factorization. Students also learn various techniques to determine cube roots using prime factorization and estimation methods. The chapter builds foundational understanding for higher concepts like exponents and volume in later classes.

II. Key Concepts Covered:

ConceptExplanation
Perfect CubeA number that is the cube of a whole number. Example: $27=3327 = 3^3
Properties of CubesCubes of even numbers are even; cubes of odd numbers are odd.
Last Digit PatternHelps in identifying whether a number is a perfect cube or not.
Cube RootA number that when cubed gives the original number.
Prime Factorization MethodUsed to find cube roots by grouping prime factors in triplets.
Estimation MethodUsed for cube roots of non-perfect cubes or large numbers.
Cube of a Negative NumberAlways negative. Example: $(−2)3=−8(-2)^3 = -8

III. Important Questions:

(A) Multiple Choice Questions (1 Mark):

  1. Cube of 0 is:
    a) 0 ✔️
    b) 1
    c) Undefined
    d) Not a number

  2. Which of the following is a perfect cube?
    a) 64 ✔️
    b) 48
    c) 36
    d) 81

  3. The cube of an even number is:
    a) Always odd
    b) Always even ✔️
    c) Sometimes even
    d) Always prime

  4. Cube root of 512 is:
    a) 8 ✔️
    b) 9
    c) 6
    d) 7

(B) Short Answer Questions (2/3 Marks):

  1. Find the cube of -5.

  2. Write the cube root of 729 using prime factorization.

  3. Find the smallest number by which 128 must be multiplied to make it a perfect cube.

  4. State whether 216 is a perfect cube. Give reason.

(C) Long Answer Questions (5 Marks):

  1. Find the cube roots of the following numbers using prime factorization:
    a) 3375
    b) 5832

  2. Is 1000 a perfect cube? Explain and find its cube root.

  3. Find the smallest number by which 72 must be multiplied to make it a perfect cube.

  4. The volume of a cube is 2197 cm³. Find the length of its side.

(D) HOTS (Higher Order Thinking Skills):

  1. Why is the cube of any odd number always odd? Prove using algebra.

  2. A number when multiplied by itself three times gives 9261. Find the number without using a calculator.

IV. Key Formulas/Concepts:

ConceptFormula / Rule
Cube of a number$a3=a×a×aa^3 = a \times a \times a
Cube root$x3\sqrt[3]{x}
Prime FactorizationGrouping prime factors in triples
Cube of negative number$(−a)3=−a3(-a)^3 = -a^3

V. Deleted Portions (CBSE 2025–2026):

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025–2026):

Unit/ChapterEstimated MarksType of Questions Typically Asked
Cubes & Cube Roots5–7 MarksCube finding, cube root via factorization, HOTS

VII. Previous Year Questions (PYQs):

MarksQuestionYear
3 MarksFind the cube root of 512 using prime factorization.2020
2 MarksIs 343 a perfect cube? Justify your answer.2021
5 MarksFind the least number to multiply 1323 to make a perfect cube.2019

VIII. Real-World Application Examples to Connect with Topics:

  • Geometry: Used in calculating the volume of cubes and cuboids.

  • Construction: Measuring and balancing cube-shaped structures or tiles.

  • Computer Science: Algorithms dealing with 3D modeling and cryptography.

  • Gaming & Design: 3D modeling of cube-shaped objects in virtual space.

IX. Student Tips & Strategies for Success (Class-Specific):

Time Management:

  • Memorize cubes of numbers from 1 to 20.

  • Solve 5 cube root questions daily for 15 minutes.

Exam Preparation:

  • Practice prime factorization method thoroughly.

  • Focus on cube-related word problems and HOTS.

Stress Management:

  • Use cube models or building blocks to visualize concepts.

  • Engage in cube root games/quizzes for fun learning.

X. Career Guidance & Exploration (Class-Specific):

For Class 9–10 Students:
StreamCareer Paths
ScienceEngineer (civil, mechanical, 3D modeling), Architect
CommerceData Analyst, Financial Modeler
ArtsVisual Designer, Game Developer
Explore:

  • NTSE, Math Olympiads, SOF International Mathematics Olympiad (IMO)

XI. Important Notes:

  • The cube of a number can be positive or negative depending on the sign.

  • Check cube pattern digits to quickly judge if a number might be a perfect cube.

  • Keep a separate cube table notebook for quick revision.

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