1. Chapter at a Glance
If you fold a sheet of paper in half again and again, its thickness doubles each time: 2, 4, 8, 16, … Starting from 0.1 mm, after about 42 folds it would be thicker than the distance from the Earth to the Moon! Repeated multiplication grows astonishingly fast, and exponents (powers) give us a short way to write it. This chapter teaches how to read and write powers, the laws of exponents, negative and zero exponents, and scientific notation for very large and very small numbers.
- Base, exponent and power; reading expressions like 2⁵.
- Laws of exponents for multiplication, division and powers of powers.
- Zero and negative exponents.
- Scientific (standard) notation.
- Comparing very large and very small quantities.
- Ancient Indian names for powers of ten.
2. Exponents and Powers
In aⁿ, a is the base and n is the exponent (or index): aⁿ = a × a × … × a (n times).
- 2⁵ = 2 × 2 × 2 × 2 × 2 = 32 (read "2 to the power 5").
- 10⁶ = 10,00,000 (one million).
- (−3)⁴ = 81 but (−3)³ = −27: an even power of a negative number is positive, an odd power negative.
Doubling grows fast
A paper about 0.1 mm thick, folded 10 times, becomes 0.1 × 2¹⁰ = 102.4 mm ≈ 10 cm thick. After 20 folds it is about 2¹⁰ times more: ≈ 105 m. After 30 folds, ≈ 107 km. This shows the power of exponential growth.
Write 729 as a power of 3, and 1024 as a power of 2 and of 4.
Solution: 729 = 3⁶. 1024 = 2¹⁰ = (2²)⁵ = 4⁵.
3. Laws of Exponents
For non-zero numbers a, b and whole numbers m, n:
| Law | Rule | Example |
|---|---|---|
| Product of powers | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
| Quotient of powers | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁸ ÷ 5³ = 5⁵ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (3²)⁴ = 3⁸ |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2 × 5)³ = 2³ × 5³ |
| Power of a quotient | (a/b)ⁿ = aⁿ/bⁿ | (2/3)⁴ = 16/81 |
| Zero exponent | a⁰ = 1 | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
Why a⁰ = 1: a³ ÷ a³ = 1, but by the quotient law it is a³⁻³ = a⁰. So a⁰ must be 1.
Why a⁻ⁿ = 1/aⁿ: a² ÷ a⁵ = 1/a³, and by the quotient law it is a⁻³.
Simplify (2⁵ × 2³) ÷ 2⁶ and (3²)³ × 3⁻⁴.
Solution: 2⁸ ÷ 2⁶ = 2² = 4. 3⁶ × 3⁻⁴ = 3² = 9.
Find x if 5ˣ⁺¹ = 125.
Solution: 125 = 5³, so x + 1 = 3 and x = 2.
4. Scientific Notation
Very large and very small numbers are written in scientific (standard) form: a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
- Distance of the Sun from the Earth ≈ 150,000,000 km = 1.5 × 10⁸ km.
- Speed of light ≈ 300,000 km/s = 3 × 10⁵ km/s.
- Thickness of a human hair ≈ 0.00008 m = 8 × 10⁻⁵ m.
- A red blood cell is about 0.000007 m = 7 × 10⁻⁶ m across.
- Place the decimal point after the first non-zero digit.
- Count how many places the point moved: that is n.
- If the original number was large, n is positive; if it was less than 1, n is negative.
Write 4,32,00,000 and 0.000608 in standard form.
Solution: 4,32,00,000 = 4.32 × 10⁷. 0.000608 = 6.08 × 10⁻⁴.
Comparing quantities
Scientific notation makes comparisons easy. The Sun's diameter (≈ 1.4 × 10⁹ m) compared with the Earth's (≈ 1.3 × 10⁷ m): 10⁹ ÷ 10⁷ = 10², so the Sun is roughly 100 times wider.
5. A Pinch of History: Indian Names for Powers of Ten
Ancient Indian texts gave names to very large powers of ten, far beyond what was needed in daily life. A Buddhist text, the Lalitavistara, describes names reaching 10⁵³ (tallakṣaṇa), and Jain texts went even further. Today's Indian names continue the pattern: lakh (10⁵), crore (10⁷), arab (10⁹), kharab (10¹¹), neel (10¹³), padma (10¹⁵), shankh (10¹⁷). The International names go thousand (10³), million (10⁶), billion (10⁹), trillion (10¹²), quadrillion (10¹⁵). A googol is 10¹⁰⁰ — larger than the estimated number of atoms in the observable universe (about 10⁷⁸–10⁸²).
6. Solved Problems (NCERT Style)
Simplify (2³ × 3⁴ × 4) ÷ (3 × 32).
Solution: 4 = 2², 32 = 2⁵. Numerator = 2⁵ × 3⁴; denominator = 3 × 2⁵. Result = 3³ = 27.
Evaluate (1/2)⁻² + (1/3)⁻² + (1/4)⁻².
Solution: (1/a)⁻² = a². So 4 + 9 + 16 = 29.
Find (3 × 10⁴) × (2.5 × 10⁻⁶) in standard form.
Solution: 3 × 2.5 = 7.5; 10⁴ × 10⁻⁶ = 10⁻². Answer = 7.5 × 10⁻².
A bacterium doubles every hour. Starting from one, how many are there after 10 hours?
Solution: 2¹⁰ = 1024.
7. Master Formula Table
| Law | Formula |
|---|---|
| Product | aᵐ · aⁿ = aᵐ⁺ⁿ |
| Quotient | aᵐ / aⁿ = aᵐ⁻ⁿ |
| Power of power | (aᵐ)ⁿ = aᵐⁿ |
| Product to a power | (ab)ⁿ = aⁿbⁿ |
| Quotient to a power | (a/b)ⁿ = aⁿ/bⁿ |
| Zero exponent | a⁰ = 1 (a ≠ 0) |
| Negative exponent | a⁻ⁿ = 1/aⁿ; (a/b)⁻ⁿ = (b/a)ⁿ |
| Standard form | a × 10ⁿ, 1 ≤ a < 10 |
| Sign of (−a)ⁿ | + if n even, − if n odd |
8. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Powers | Repeated multiplication | Expand and evaluate |
| Laws | Add, subtract, multiply exponents | Simplify expressions |
| Zero/negative | a⁰ = 1, a⁻ⁿ = 1/aⁿ | Evaluate |
| Standard form | a × 10ⁿ | Convert, compare |
| Growth | Doubling is explosive | Word problems |
9. Mind Map
- Base, exponent
- Signs of powers
- Product, quotient
- Power of power
- Zero, negative
- a × 10ⁿ
- Big and tiny numbers
- Paper folding
- Indian number names
- Googol
10. Tips and Tricks
- Convert all bases to primes first (4 = 2², 27 = 3³), then apply the laws.
- To compare 2³⁰ and 3²⁰: write as (2³)¹⁰ = 8¹⁰ and (3²)¹⁰ = 9¹⁰ — so 3²⁰ is bigger.
- Remember 2¹⁰ = 1024 ≈ 10³.
- Multiplying exponents when bases are multiplied: 2³ × 2⁴ is 2⁷, not 2¹².
- Thinking a negative exponent makes the number negative: 2⁻³ = 1/8, not −8.
- Writing 45 × 10⁶ as standard form (should be 4.5 × 10⁷).
- Adding powers: 2³ + 2³ = 2 × 2³ = 2⁴, not 2⁶.
11. Important One-liners
- aᵐ × aⁿ = aᵐ⁺ⁿ.
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
- (aᵐ)ⁿ = aᵐⁿ.
- Any non-zero number to the power 0 is 1.
- a⁻ⁿ is the reciprocal of aⁿ.
- In standard form, the number before × 10ⁿ lies between 1 and 10.
- 2¹⁰ = 1024.
- A googol is 10¹⁰⁰.
12. Exam-pattern Questions with Detailed Answers
The value of 3⁴ is:
- 12
- 64
- 81
- 27
Show answer
3 × 3 × 3 × 3 = 81.
Answer: (C) 81
Simplify 7⁵ ÷ 7³.
Show answer
7⁵⁻³ = 7² = 49.
Answer: 49
Write 0.00045 in standard form.
Show answer
Move the point 4 places right: 4.5 × 10⁻⁴.
Answer: 4.5 × 10⁻⁴
Evaluate (2⁻¹ + 3⁻¹)⁻¹.
Show answer
½ + ⅓ = 5/6; its reciprocal is 6/5.
Answer: 6/5
Find m if 2ᵐ × 2⁵ = 2¹².
Show answer
m + 5 = 12 → m = 7.
Answer: m = 7
Which is greater: 2⁴⁰ or 4²⁰?
- 2⁴⁰
- 4²⁰
- Equal
- Cannot compare
Show answer
4²⁰ = (2²)²⁰ = 2⁴⁰. They are equal.
Answer: (C) Equal
Simplify (25 × t⁻⁴) ÷ (5⁻³ × 10 × t⁻⁸), t ≠ 0.
Show answer
= 5² × 5³ × t⁻⁴⁺⁸ ÷ 10 = 5⁵ t⁴ ÷ 10 = 3125 t⁴ / 10 = 625t⁴/2.
Answer: 625t⁴/2
Express 3.07 × 10⁶ in usual form.
Show answer
Move the point 6 places right: 3,070,000 (30,70,000).
Answer: 30,70,000
Compare 2³⁰ and 3²⁰ without calculating them fully.
Show answer
2³⁰ = (2³)¹⁰ = 8¹⁰ and 3²⁰ = (3²)¹⁰ = 9¹⁰. Since 9 > 8, 3²⁰ is greater.
Answer: 3²⁰ > 2³⁰
Solve (3/5)ˣ × (5/3)⁴ = (3/5)⁷.
Show answer
(5/3)⁴ = (3/5)⁻⁴. So x − 4 = 7 → x = 11.
Answer: x = 11
The mass of the Earth is about 5.97 × 10²⁴ kg and that of the Moon about 7.35 × 10²² kg. About how many times heavier is the Earth?
Show answer
5.97 × 10²⁴ ÷ 7.35 × 10²² ≈ (5.97 ÷ 7.35) × 10² ≈ 0.81 × 100 ≈ 81.
Answer: About 81 times
A sheet of paper is 0.1 mm thick. After how many folds would its thickness first exceed 1 m? (Assume folding is possible.)
Show answer
Thickness after n folds = 0.1 × 2ⁿ mm. Need > 1000 mm → 2ⁿ > 10,000. 2¹³ = 8192, 2¹⁴ = 16,384.
Answer: 14 folds
13. Quick Revision Summary
- aⁿ = a multiplied n times; even powers of negatives are positive.
- Laws: add exponents (×), subtract (÷), multiply ((aᵐ)ⁿ); a⁰ = 1; a⁻ⁿ = 1/aⁿ.
- Standard form a × 10ⁿ with 1 ≤ a < 10.
- Compare powers by rewriting with a common base or common exponent.
- Indian texts named powers of ten up to 10⁵³ and beyond.