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Power Play – Class 8 Maths Chapter 2 Notes & NCERT PDF

Class 8MathematicsChapter 2NCERT book: Ganita Prakash - Mathematics Class 8

Notes and a simple summary of Chapter 2 of the NCERT Class 8 Mathematics book, with the official chapter PDF, flashcards, an MCQ quiz and an AI tutor for your doubts.

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Chapter 2: Power Play

Introduction

The chapter starts with An Impossible Venture: how many times can you fold a sheet of paper? Suppose a sheet is 0.001 cm thick and every fold doubles the thickness.

  • After 10 folds it is just over 1 cm (1.024 cm).
  • After 17 folds it is about 131 cm.
  • After 30 folds it is about 10.7 km, the height at which planes fly.
  • After 46 folds it would be more than 7,00,000 km — enough to reach the Moon!

    • This is exponential (multiplicative) growth. Every 10 folds multiplies the thickness by 1024 (2 multiplied by itself 10 times).

    Key Ideas

    2.2 Exponential Notation

  • nᵃ means n multiplied by itself a times. In 5⁴ = 5 × 5 × 5 × 5 = 625, 5 is the base and 4 is the exponent (power); 5⁴ is read "5 raised to the power 4".
  • The thickness after 10 folds is 2¹⁰v, where v is the starting thickness.
  • Negative bases: (−4)³ = −64; (−1) to an odd power is −1 and to an even power is +1.
  • Products of different letters: a × a × a × b × b = a³b².
  • Remember: 4 + 4 + 4 = 3 × 4 = 12, but 4 × 4 × 4 = 4³ = 64.
  • Prime factors in exponential form: 32400 = 2⁴ × 5² × 3⁴.

    • The Laws of Exponents

    nᵃ × nᵇ = nᵃ⁺ᵇ
    (nᵃ)ᵇ = (nᵇ)ᵃ = nᵃᵇ
    mᵃ × nᵃ = (mn)ᵃ and mᵃ ÷ nᵃ = (m ÷ n)ᵃ
    nᵃ ÷ nᵇ = nᵃ⁻ᵇ (n ≠ 0)
    n⁰ = 1 and n⁻ᵃ = 1/nᵃ (n ≠ 0)
  • The Stones that Shine: 3 daughters → 3 baskets each → 3 keys → 3 rooms gives 3⁴ = 81 rooms; with 3 tables, 3 necklaces and 3 diamonds each, there are 3⁷ = 2187 diamonds, which is 3⁴ × 3³ = 81 × 27.
  • Magical Pond: lotuses double daily and cover the pond on day 30, so the pond was half full on day 29. Doubling for 4 days then tripling for 4 days gives 2⁴ × 3⁴ = (2 × 3)⁴ = 6⁴ = 1296, in either order.
  • The other side of powers: halving a line of 2⁴ = 16 units shows 2⁴ ÷ 2³ = 2¹. Defining 2⁰ = 2⁴ ÷ 2⁴ = 1 keeps the rule working; halving more times than possible gives negative exponents, e.g. 2⁴ ÷ 2⁵ = 2⁻¹ = ½.
  • Power lines: on a line of powers of 4, moving one step right multiplies by 4; 4⁷ is 4² = 16 times larger than 4⁵.

    • How Many Combinations

  • Estu with 4 dresses and 3 caps has 4 × 3 = 12 outfits; Roxie with 7 dresses, 2 hats and 3 pairs of shoes has 7 × 2 × 3 = 42.
  • A 5-digit lock (0–9) has 10 × 10 × 10 × 10 × 10 = 10⁵ = 1,00,000 passwords; a 6-slot lock with letters A–Z has 26⁶.

    • 2.4 Powers of 10 and Scientific Notation

  • Expanded form with powers of 10: 47561 = 4 × 10⁴ + 7 × 10³ + 5 × 10² + 6 × 10¹ + 1 × 10⁰; decimals use negative powers: 561.903 = … + 9 × 10⁻¹ + 0 × 10⁻² + 3 × 10⁻³.
  • Scientific (standard) form: x × 10ʸ, where 1 ≤ x < 10 and y is an integer. 5900 = 5.9 × 10³; 80,00,000 = 8 × 10⁶.
  • The exponent matters most: changing Mumbai's 2 × 10⁷ to 3 × 10⁷ adds half again, but changing it to 2 × 10⁸ makes it ten times bigger.
  • The number of digits in the coefficient shows how precisely we know the number (Kohima ≈ 1.42 × 10⁵).
  • Sun–Earth 1.496 × 10¹¹ m is the smallest of the three distances compared with Sun–Saturn (1.4335 × 10¹² m) and Saturn–Uranus (1.439 × 10¹² m).

    • 2.5 Did You Ever Wonder?

  • Solve estimation questions by guessing first, then describing the relationships, making reasonable assumptions and calculating. Examples: Tulābhāra (offering goods equal to a person's weight) and pādayātra.
  • Linear vs exponential growth: a ladder to the Moon (3,84,400 km) with 20 cm steps needs 1,92,20,00,000 steps (linear, additive), while paper reaches it in 46 folds (exponential, multiplicative).
  • Indian and international names: lakh 10⁵, crore 10⁷, arab 10⁹, kharab 10¹¹, neel 10¹³, padma 10¹⁵, shankh 10¹⁷, maha shankh 10¹⁹; million 10⁶, billion 10⁹, trillion 10¹², quadrillion 10¹⁵.
  • Time in seconds: 10⁶ s ≈ 11.57 days, but 10⁹ s ≈ 31.7 years. A googol is 10¹⁰⁰.

    • Worked Examples

  • Compute 5⁷ using powers you know. 5⁷ = 5³ × 5⁴ = 125 × 625 = 78125.
  • Write 8⁶ as a power of a power. 8⁶ = (8²)³, and since 8 = 2³, 8⁶ = (2³)⁶ = 2¹⁸.
  • Simplify 2¹⁰⁰ ÷ 2²⁵. Subtract exponents: 2⁷⁵.
  • Express 648 in prime-power form. 648 = 2 × 2 × 2 × 3 × 3 × 3 × 3 = 2³ × 3⁴.
  • Units digit of 2²²⁴ ÷ 4³². 4³² = (2²)³² = 2⁶⁴, so the quotient is 2¹⁶⁰; powers of 2 with exponents that are multiples of 4 end in 6, so the units digit is 6.
  • Which is greater, 2¹⁰⁰ or 100²? 100² = 10⁴, but 2¹⁰⁰ = (2¹⁰)¹⁰ > (10³)¹⁰ = 10³⁰, so 2¹⁰⁰ is far greater.

    • Figure it Out — Hints

  • Write each number as prime powers before comparing (2⁴ × 3⁶ = 6⁴ × 3² = 18² × 6²).
  • 4³ = 64 < 81 = 3⁴, and 2⁸ = 256 > 64 = 8².
  • For the 8.5 billion milk packets: 10⁹ codes are too few, so the code needs 10 digits.
  • Numbers that are both squares and cubes are sixth powers (1, 64, 729, …).
  • 10⁹ + 10⁹ = 2 × 10⁹, not 10¹⁸.

    • Maths Words

  • Exponent / power (घातांक)
  • Base (आधार)
  • Exponential form (घातांकीय रूप)
  • Laws of exponents (घातांक के नियम)
  • Reciprocal (व्युत्क्रम)
  • Scientific notation / standard form (वैज्ञानिक संकेतन / मानक रूप)
  • Coefficient (गुणांक)
  • Exponential growth (घातीय वृद्धि)
  • Linear growth (रैखिक वृद्धि)
  • Estimate / approximation (अनुमान / सन्निकटन)
  • Combination (संयोजन)
  • Prime factorisation (अभाज्य गुणनखंडन)
  • Negative exponent (ऋणात्मक घातांक)

    • A Pinch of History

  • The Lalitavistara (a Buddhist text, 1st century BCE) names odd powers of ten up to 10⁵³ (tallakshana), e.g. ayuta = 10⁹ and niyuta = 10¹¹.
  • Mahaviracharya listed 24 terms (up to 10²³) in the Ganita-sara-sangraha; the Jaina text Amalasiddhi names powers up to 10⁹⁶; Kāccāyana's Pali grammar goes up to 10¹⁴⁰ (asaṅkhyeya).
  • The Indian naming pattern (a hundred thousand is a lakh, a hundred lakhs a crore) follows the same idea as the old texts.

    • Common Mistakes

  • Treating 2³ as 2 × 3 = 6 instead of 2 × 2 × 2 = 8.
  • Adding exponents when bases differ (2³ × 3² is not 6⁵).
  • Multiplying exponents in nᵃ × nᵇ — they should be added.
  • Thinking n⁰ = 0 (it is 1) or that n⁻ᵃ is negative (it is 1/nᵃ).
  • Writing scientific notation with a coefficient of 10 or more (59 × 10² should be 5.9 × 10³).

    • Maths in Daily Life

    Powers of 10 are used for population, distances in space and ages of fossils; exponents count passwords, PIN codes and vehicle numbers. Understanding exponential growth helps us see why things like viral messages or compound growth get huge very quickly.

    💡 Key Learning Points

    • ✓nᵃ means n multiplied by itself a times — the base is n and the exponent is a
    • ✓Laws of exponents: nᵃ × nᵇ = nᵃ⁺ᵇ, (nᵃ)ᵇ = nᵃᵇ, nᵃ ÷ nᵇ = nᵃ⁻ᵇ, mᵃ × nᵃ = (mn)ᵃ
    • ✓n⁰ = 1 and n⁻ᵃ = 1/nᵃ for n ≠ 0
    • ✓Scientific notation writes numbers as x × 10ʸ with 1 ≤ x < 10; the exponent shows the size
    • ✓Exponential (multiplicative) growth is far faster than linear (additive) growth — 46 paper folds reach the Moon

    👨‍🏫 Teaching Tips

    • →Start by having students actually fold paper and record how many folds they manage, then build the thickness table together
    • →Draw the 'Stones that Shine' tree on the board so students see each level multiply by 3
    • →Let students discover the laws by expanding products such as p⁴ × p⁶ into repeated multiplication
    • →Use a paper strip halved repeatedly to show why 2⁰ = 1 and why negative exponents give fractions
    • →Make a 'power line' of 4s or 7s on the floor and have students hop left and right to multiply and divide
    • →Collect real large-number facts (ants, trees, stars) and have groups rewrite them in scientific notation
    • →Run the 'Tremendous in Ten' game so students compete to write the largest number using exponents

    📋 Assessment Questions

    1. Can the student write a product like a × a × a × c × c × c × c × d in exponential form (a³c⁴d)?
    2. Can the student apply the laws of exponents to simplify expressions like 3² × 3⁻⁵ × 3⁶?
    3. Can the student explain why 2⁰ = 1 and what 2⁻⁴ means?
    4. Can the student convert numbers such as 70,04,00,00,000 to scientific notation?
    5. Can the student compare linear and exponential growth using the ladder-to-the-Moon and paper-folding examples?

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    All chapters in Ganita Prakash - Mathematics Class 8

    1. 1. A Square and a Cube
    2. 2. Power Play
    3. 3. A Story of Numbers
    4. 4. Quadrilaterals
    5. 5. Number Play
    6. 6. We Distribute, Yet Things Multiply
    7. 7. Proportional Reasoning-1