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NCERT Class 8 · Mathematics Based on NCERT Class 8 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter explores the concept of exponential growth through the fascinating example of folding a paper to reach the Moon. It introduces exponential notation, laws of exponents (multiplication, division, power of a power), and the idea of negative and zero exponents. The big idea is that multiplicative (exponential) growth is incredibly powerful and much faster than additive (linear) growth, and we use exponents to express very large and very small numbers concisely.
Key Concepts (12)
Exponential Growth
Growth in which a quantity increases by a constant factor (multiplied) over equal intervals of time or steps.
e.g. Folding a paper doubles its thickness each time; after 46 folds, it reaches the Moon.
Base
The number that is multiplied by itself in an exponential expression.
e.g. In 5⁴, the base is 5.
Exponent (Power)
The number that tells how many times the base is used as a factor in multiplication.
e.g. In 5⁴, the exponent is 4, meaning 5 × 5 × 5 × 5.
Exponential Form
A way of writing a number as a base raised to an exponent.
e.g. 625 written in exponential form is 5⁴.
n⁰ (Zero Exponent)
Any non-zero number raised to the power 0 equals 1.
e.g. 2⁰ = 1, 10⁰ = 1, x⁰ = 1 (where x ≠ 0).
n⁻ᵃ (Negative Exponent)
A negative exponent means the reciprocal of the base raised to the positive exponent.
e.g. 2⁻¹ = 1/2, 10⁻³ = 1/1000.
Scientific Notation (Standard Form)
A way of writing numbers as x × 10ʸ, where 1 ≤ x < 10 and y is an integer.
e.g. Distance from Sun to Earth is 1.496 × 10¹¹ m.
Coefficient (in Scientific Notation)
The number x in the scientific notation x × 10ʸ, which is between 1 and 10.
e.g. In 5.9 × 10³, the coefficient is 5.9.
Linear Growth
Growth in which a quantity increases by a constant amount (addition) over equal intervals.
e.g. Climbing a ladder: each step adds a fixed 20 cm to your height.
Multiplicative Growth
Another name for exponential growth, where the quantity is multiplied by a constant factor each time.
e.g. Doubling of lotuses in a pond every day.
Googol
The number 10¹⁰⁰ (1 followed by 100 zeros).
e.g. A googol is 10¹⁰⁰.
Googolplex
The number 10 raised to the power of a googol (10^(10¹⁰⁰)).
e.g. A googolplex is 10^(googol).
Formulas & Equations (7)
nᵃ × nᵇ = nᵃ⁺ᵇ
Reads as: n raised to the power a times n raised to the power b equals n raised to the power a plus b
Variables: n = base (any non-zero number), a, b = exponents (counting numbers or integers)
Use when: Multiplying two powers with the same base
⚠ Students multiply the exponents instead of adding them, e.g., writing 2³ × 2⁴ = 2¹² instead of 2⁷.
nᵃ ÷ nᵇ = nᵃ⁻ᵇ
Reads as: n raised to the power a divided by n raised to the power b equals n raised to the power a minus b
Variables: n = base (n ≠ 0), a, b = exponents (a > b for positive result, but works for all integers)
Use when: Dividing two powers with the same base
⚠ Students divide the exponents instead of subtracting, e.g., writing 2⁶ ÷ 2² = 2³ instead of 2⁴.
(nᵃ)ᵇ = nᵃᵇ
Reads as: n raised to the power a, all raised to the power b, equals n raised to the power a times b
Variables: n = base, a, b = exponents (counting numbers)
Use when: Raising a power to another power
⚠ Students add the exponents instead of multiplying, e.g., writing (2³)² = 2⁵ instead of 2⁶.
mᵃ × nᵃ = (m × n)ᵃ
Reads as: m raised to the power a times n raised to the power a equals (m times n) raised to the power a
Variables: m, n = bases, a = exponent (counting number)
Use when: Multiplying two powers with the same exponent but different bases
⚠ Students multiply the bases and also multiply the exponents, e.g., writing 2³ × 3³ = 6⁹ instead of 6³.
mᵃ ÷ nᵃ = (m/n)ᵃ
Reads as: m raised to the power a divided by n raised to the power a equals (m divided by n) raised to the power a
Variables: m, n = bases (n ≠ 0), a = exponent (counting number)
Use when: Dividing two powers with the same exponent but different bases
⚠ Students divide the exponents instead of keeping them the same, e.g., writing 10⁴ ÷ 5⁴ = 2⁰ instead of 2⁴.
n⁰ = 1 (n ≠ 0)
Reads as: Any non-zero number raised to the power zero equals one
Variables: n = base (n ≠ 0)
Use when: Simplifying expressions where the exponent becomes zero after subtraction
⚠ Students think any number to the power 0 is 0, e.g., writing 5⁰ = 0 instead of 1.
n⁻ᵃ = 1/nᵃ (n ≠ 0)
Reads as: n raised to the power negative a equals 1 divided by n raised to the power a
Variables: n = base (n ≠ 0), a = positive integer
Use when: Rewriting expressions with negative exponents as positive exponents
⚠ Students think a negative exponent makes the result negative, e.g., writing 2⁻³ = -8 instead of 1/8.
How It Works
Paper Folding (Exponential Growth Demonstration)
  1. Start with a sheet of paper of thickness 0.001 cm.
  2. Fold it once: thickness becomes 0.001 × 2 = 0.002 cm.
  3. Fold it again: thickness becomes 0.001 × 2 × 2 = 0.004 cm.
  4. Continue folding; after each fold, thickness doubles (multiplied by 2).
  5. After n folds, thickness = initial thickness × 2ⁿ.
  6. After 46 folds, thickness exceeds 7,00,000 km (reaches the Moon).
💡 The thickness doubles each time, not adds. This is multiplicative (exponential) growth, not additive (linear).
Writing a Number in Exponential Form from Prime Factors
  1. Find the prime factors of the number using a factor tree or division method.
  2. Group the same prime factors together.
  3. Count how many times each prime appears; that count becomes the exponent.
  4. Write the product of primes raised to their respective exponents.
💡 The exponent tells how many times the base is multiplied by itself, not how many times it is added.
Using Laws of Exponents to Simplify Expressions
  1. Identify if the bases are the same or different.
  2. If multiplying same bases, add the exponents (nᵃ × nᵇ = nᵃ⁺ᵇ).
  3. If dividing same bases, subtract the exponents (nᵃ ÷ nᵇ = nᵃ⁻ᵇ).
  4. If raising a power to a power, multiply the exponents ((nᵃ)ᵇ = nᵃᵇ).
  5. If multiplying different bases with same exponent, multiply the bases and keep the exponent (mᵃ × nᵃ = (m×n)ᵃ).
  6. If dividing different bases with same exponent, divide the bases and keep the exponent (mᵃ ÷ nᵃ = (m/n)ᵃ).
💡 These laws only work when the bases are the same (for addition/subtraction of exponents) or when the exponents are the same (for multiplication/division of bases).
Converting a Number to Scientific Notation
  1. Place the decimal point after the first non-zero digit to get a number between 1 and 10.
  2. Count how many places the decimal point moved. This count is the exponent.
  3. If the decimal moved left, the exponent is positive. If it moved right, the exponent is negative.
  4. Write the number as (new number) × 10^(exponent).
💡 The coefficient must be between 1 and 10 (including 1 but not 10). For example, 5900 becomes 5.9 × 10³, not 59 × 10².
Finding Number of Combinations (Fundamental Counting Principle)
  1. Identify the number of choices for each independent event or item.
  2. Multiply the number of choices for all events together.
  3. The product gives the total number of possible combinations.
💡 This is an application of exponential growth: for a 5-digit lock, each digit has 10 choices, so total passwords = 10⁵ = 1,00,000.
Exam Traps — Don't Make These Mistakes
⚠ Confusing Addition with Multiplication in Exponents
✗ 2³ × 2⁴ = 2¹² (multiplying exponents instead of adding)
→ The law says nᵃ × nᵇ = nᵃ⁺ᵇ, not nᵃ×ᵇ. Multiplication of powers with same base requires adding exponents.
✓ 2³ × 2⁴ = 2³⁺⁴ = 2⁷ = 128
⚠ Negative Exponent Means Negative Number
✗ 2⁻³ = -8
→ A negative exponent does not make the result negative; it means the reciprocal of the positive power.
✓ 2⁻³ = 1/2³ = 1/8
⚠ Zero Exponent Equals Zero
✗ 5⁰ = 0
→ Any non-zero number raised to the power 0 is 1, not 0. This comes from nᵃ ÷ nᵃ = n⁰ = 1.
✓ 5⁰ = 1
⚠ Incorrectly Applying Power of a Power
✗ (2³)² = 2⁵ (adding exponents instead of multiplying)
→ The law says (nᵃ)ᵇ = nᵃᵇ, so exponents must be multiplied, not added.
✓ (2³)² = 2³×² = 2⁶ = 64
⚠ Mixing Up Linear and Exponential Growth
✗ After 46 folds, thickness = 0.001 × 46 = 0.046 cm
→ Folding multiplies thickness by 2 each time (exponential), not adds 0.001 cm each time (linear).
✓ After 46 folds, thickness = 0.001 × 2⁴⁶ cm, which is over 7,00,000 km.
⚠ Scientific Notation Coefficient Not Between 1 and 10
✗ 5900 = 59 × 10²
→ In scientific notation, the coefficient must be between 1 and 10. 59 is not between 1 and 10.
✓ 5900 = 5.9 × 10³
Compare & Contrast
How the quantity changes over equal steps
Exponential Growth
Linear Growth
Exponential growth multiplies by a constant factor each step (e.g., doubling), while linear growth adds a constant amount each step (e.g., adding 20 cm).
Meaning and value of the expression
Positive Exponent
Negative Exponent
A positive exponent (e.g., 2³) means repeated multiplication (2 × 2 × 2 = 8), while a negative exponent (e.g., 2⁻³) means the reciprocal of the positive power (1/2³ = 1/8).
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 Very Short Answer (1 mark), 1 Short Answer (3 marks), 1 Long Answer (5 marks) — total approximately 10-12 marks
Most asked: Laws of exponents (multiplication, division, power of a power) · Expressing numbers in exponential form using prime factors · Scientific notation (standard form) for large and small numbers · Zero and negative exponents
Don't skip: Simplify expressions using multiple laws of exponents (e.g., 2⁻⁴ × 2⁷ ÷ 2³); Convert a large number (e.g., 70,04,00,00,000) to scientific notation; Word problems involving exponential growth (e.g., paper folding, lotus doubling)
One-Line Revision (read aloud before the exam)
  1. Exponential growth multiplies by a constant factor each step; linear growth adds a constant amount.
  2. In exponential form nᵃ, n is the base and a is the exponent (power).
  3. nᵃ × nᵇ = nᵃ⁺ᵇ (add exponents when multiplying same bases).
  4. nᵃ ÷ nᵇ = nᵃ⁻ᵇ (subtract exponents when dividing same bases).
  5. (nᵃ)ᵇ = nᵃᵇ (multiply exponents when raising a power to a power).
  6. mᵃ × nᵃ = (m × n)ᵃ (multiply bases when exponents are same).
  7. mᵃ ÷ nᵃ = (m/n)ᵃ (divide bases when exponents are same).
  8. Any non-zero number raised to the power 0 equals 1 (n⁰ = 1, n ≠ 0).
  9. A negative exponent means reciprocal: n⁻ᵃ = 1/nᵃ (n ≠ 0).
  10. Scientific notation: write a number as x × 10ʸ, where 1 ≤ x < 10 and y is an integer.
  11. After 46 folds, a paper of thickness 0.001 cm reaches the Moon (over 7,00,000 km).
  12. A 5-digit lock has 10⁵ = 1,00,000 possible passwords.
  13. A googol is 10¹⁰⁰; a googolplex is 10^(googol).
  14. Prime factors of 32400 in exponential form: 2⁴ × 3⁴ × 5².

MCQs — 10 Questions with Answers

Q1. What is 3^4?

  • A. 12
  • B. 81 ✓
  • C. 27
  • D. 64

Answer: B — 3^4 means 3 × 3 × 3 × 3 = 9 × 9 = 81; students often confuse it with 3 × 4 = 12.

Q2. Express 2 × 2 × 2 × 5 × 5 in exponential form.

  • A. 2^5 × 5^2
  • B. 2^3 × 5^2 ✓
  • C. 2^2 × 5^3
  • D. (2 × 5)^5

Answer: B — Count: three 2's and two 5's, so 2^3 × 5^2; students may recount or confuse which base has which exponent.

Q3. What is the value of 5^0?

  • A. 0
  • B. 1 ✓
  • C. 5
  • D. undefined

Answer: B — Any non-zero number to the power 0 equals 1; students often incorrectly think 5^0 = 0 or 5.

Q4.

  • A. 2^7 ✓
  • B. 2^12
  • C. 4^7
  • D. 2^1

Answer: A — When multiplying powers with the same base, add exponents: 2^4 × 2^3 = 2^(4+3) = 2^7 = 128.

Q5. A sheet of paper has thickness 0.001 cm. After folding 5 times (thickness doubles each fold), what is its thickness in exponential form?

  • A. 0.001 × 5^2 cm
  • B. 0.001 × 2^5 cm ✓
  • C. 0.001 × 10 cm
  • D. 0.05 cm

Answer: B — The thickness after 5 folds is 0.001 × 2 × 2 × 2 × 2 × 2 = 0.001 × 2^5 cm; students may confuse the exponent with the fold number or multiply incorrectly.

Q6. Roxie has 7 dresses, 2 hats, and 3 pairs of shoes. In how many ways can she dress up?

  • A. 7 + 2 + 3 = 12 ways
  • B. 7 × 2 × 3 = 42 ways ✓
  • C. 7^3 = 343 ways
  • D. 2^7 = 128 ways

Answer: B — For each dress (7 choices), she picks a hat (2 choices) and shoes (3 choices), so 7 × 2 × 3 = 42; students may add instead of multiply.

Q7. Simplify (3^2)^3 using the law of exponents.

  • A. 3^5
  • B. 3^6 ✓
  • C. 6^3
  • D. 9^3

Answer: B — Power of a power rule: multiply exponents (3^2)^3 = 3^(2×3) = 3^6 = 729; students may add exponents instead of multiplying.

Q8. A lotus pond doubles every day. If it is completely covered on day 30, on which day was it half covered?

  • A. Day 15
  • B. Day 29 ✓
  • C. Day 1
  • D. Day 28

Answer: B — Since the number doubles daily, day 29 must have half the lotuses of day 30; students may incorrectly divide 30 by 2 to get day 15.

Q9. What is 2^3 × 5^3 in simplest form?

  • A. 10^3 ✓
  • B. 10^6
  • C. 7^3
  • D. 10^9

Answer: A — Using m^a × n^a = (mn)^a: 2^3 × 5^3 = (2 × 5)^3 = 10^3 = 1000; students may add exponents or multiply bases without combining the exponents.

Q10. Express 32400 as a product of its prime factors in exponential form.

  • A. 2^3 × 3^4 × 5^2
  • B. 2^4 × 3^4 × 5^2 ✓
  • C. 2^5 × 3^3 × 5^2
  • D. 2^2 × 3^5 × 5^2

Answer: B — Prime factorisation: 32400 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 5 × 5 = 2^4 × 3^4 × 5^2; students may miscount the factors during division.

Flashcards

What does 5^4 mean in words?

5 multiplied by itself 4 times, or 5 × 5 × 5 × 5 = 625.

In the expression 7^3, what is the base and what is the exponent?

Base is 7, exponent (or power) is 3.

What is the rule for multiplying powers with the same base?

Add the exponents: n^a × n^b = n^(a+b).

What is the rule for a power raised to another power?

Multiply the exponents: (n^a)^b = n^(ab).

What is the value of any non-zero number to the power 0?

Any non-zero number to the power 0 equals 1 (e.g., 5^0 = 1).

How does a lotus pond with doubling growth relate to the answer from day 29 to day 30?

If the pond is full on day 30, it is exactly half full on day 29 because the quantity doubles each day.

What is 2^10 equal to?

2^10 = 1024, which represents the thickness increase factor every 10 paper folds.

Express 6 × 6 × 6 × a × a in exponential form.

6^3 × a^2 or 6^3a^2.

What does the sign '≈' mean in expressions like 131 cm ≈ thickness after 17 folds?

The sign '≈' means 'approximately equal to' when the exact value is rounded.

In a 5-digit password lock where each digit can be 0–9, how many total passwords are possible?

10^5 = 100,000 passwords (10 choices for each of 5 positions).

Important Board Questions

Define exponential notation and give one example with your own numbers. [1 mark]

Write n^a = n × n × ... (a times). Example: 4^3 = 4 × 4 × 4 = 64.

A paper sheet has thickness 0.001 cm. After 10 folds (doubling each fold), what is its thickness? Express the answer in exponential form and calculate the numerical value. [2 marks]

Thickness = 0.001 × 2^10 cm. Calculate 2^10 = 1024, so thickness = 0.001 × 1024 = 1.024 cm.

Simplify the following using laws of exponents and show each step: (i) 3^4 × 3^2 (ii) (5^2)^3 (iii) 2^4 × 3^4 [3 marks]

(i) Use n^a × n^b = n^(a+b): 3^4 × 3^2 = 3^6. (ii) Use (n^a)^b = n^(ab): (5^2)^3 = 5^6. (iii) Use m^a × n^a = (mn)^a: 2^4 × 3^4 = 6^4.

Estu has a safe with a 3-digit password lock. Each digit can be any number from 0 to 9. (a) Express the total number of possible passwords in exponential form. (b) Calculate the total number of passwords. (c) If a lotus pond triples its flowers every day and started with 1 lotus, express the number of lotuses after 5 days in exponential form. (d) How does exponential growth in the password combinations compare to the lotus pond growth? [5 marks]

(a) For 3 positions, each with 10 choices: 10^3. (b) 10^3 = 1000 passwords. (c) 1 × 3^5 = 3^5 = 243 lotuses. (d) Exponential growth means the quantity increases by a constant factor each step; lotus triples (factor 3) while passwords multiply by 10 per digit added.

Assertion-Reason Questions for Power Play

Q1. Assertion (A):

The exponential form 2¹⁰ equals 1024.

Reason (R):

2¹⁰ means 2 is multiplied by itself 10 times, and this product equals 1024.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. 2¹⁰ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 1024, and the reason correctly explains why this is the case by referencing the definition of exponents.

Q2. Assertion (A):

When a sheet of paper is folded 26 times starting with thickness 0.001 cm, its thickness reaches approximately 670 metres, which is close to the height of Burj Khalifa in Dubai (830 m).

Reason (R):

With each fold, the thickness doubles, creating exponential growth, and after 26 folds the thickness increases dramatically due to this multiplicative pattern.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — The assertion is true—after 26 folds, the thickness is approximately 670 m. The reason is also true, but it does not directly explain why the thickness is 670 m specifically; it only explains the general pattern of exponential growth. A complete explanation would need to show the calculation 0.001 × 2²⁶ ≈ 670 m.

Q3. Assertion (A):

The expression 3² × 3⁵ can be simplified to 3¹⁰.

Reason (R):

When multiplying powers with the same base, we multiply the exponents together.

Answer: D — A is false but R is true — The assertion is false; 3² × 3⁵ = 3⁷, not 3¹⁰. However, the reason is also false because the correct rule is to ADD exponents when multiplying powers of the same base, not multiply them. Students who know the rule would still get the assertion wrong due to incorrect application.

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Case-Based Questions (CBQ) for Power Play

Case Study:

A biologist studying bacterial growth placed 2 bacteria in a petri dish. Every 2 hours, each bacterium divides into 2, effectively doubling the population. The biologist recorded the population at 2-hour intervals: at hour 0 (start) there were 2 bacteria, at hour 2 there were 4 bacteria, at hour 4 there were 8 bacteria, at hour 6 there were 16 bacteria. The biologist wants to predict the population after 12 hours without doing manual counting.

Q1. What is the pattern in bacterial population growth?

  • A. Population increases by 2 each time period
  • B. Population doubles (multiplied by 2) every 2 hours ✓
  • C. Population increases by a fixed number each hour
  • D. Population triples every hour

Answer: B

Q2. How many 2-hour periods are there in 12 hours?

  • A. 6 periods ✓
  • B. 12 periods
  • C. 2 periods
  • D. 24 periods

Answer: A

Q3. Express the population after 12 hours using exponential notation, starting with 2 bacteria.

  • A. 2 × 2⁶ = 128 bacteria ✓
  • B. 2⁶ = 64 bacteria
  • C. 2 × 6 = 12 bacteria
  • D. 2 + 6 = 8 bacteria

Answer: A

Q4. After how many hours will the population reach 256 bacteria?

  • A. 10 hours
  • B. 12 hours
  • C. 16 hours ✓
  • D. 8 hours

Answer: C

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