Exponents and Powers — Class 8 Mathematics
1. What Are Exponents?
An exponent tells how many times a number is MULTIPLIED by itself. In aⁿ: a = BASE, n = EXPONENT/POWER. aⁿ = a × a × a × ... (n times). Examples: 2⁵ = 2×2×2×2×2 = 32. (−3)⁴ = (−3)(−3)(−3)(−3) = 81. −3⁴ = −(3⁴) = −81 (parentheses MATTER).
2. Laws of Exponents — Complete
| Law | Statement | Example |
|---|---|---|
| Product Law | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³×2⁴ = 2⁷ = 128 |
| Quotient Law | aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a≠0) | 2⁵/2² = 2³ = 8 |
| Power of Power | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ = 64 |
| Power of Product | (ab)ᵐ = aᵐ × bᵐ | (2×3)² = 4×9 = 36 |
| Power of Quotient | (a/b)ᵐ = aᵐ/bᵐ (b≠0) | (2/3)² = 4/9 |
| Zero Exponent | a⁰ = 1 (a≠0) | 7⁰ = 1, (−5)⁰ = 1 |
| Negative Exponent | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional Exponent | a^(1/n) = ⁿ√a | 8^(1/3) = ³√8 = 2 |
Deep Dive — Why a⁰ = 1?
Consider the pattern: 2³=8, 2²=4, 2¹=2, 2⁰=? Each step divides by 2. 2¹/2 = 2/2 = 1. So 2⁰ = 1. 'ANY non-zero number raised to power 0 equals 1. This is not a convention — it follows LOGICALLY from the quotient law: aᵐ/aᵐ = aᵐ⁻ᵐ = a⁰ = 1.'
Deep Dive — Negative Exponents
a⁻ⁿ = 1/aⁿ. So 2⁻³ = 1/2³ = 1/8. 'A negative exponent does NOT make the number negative. It places the number in the DENOMINATOR.' (−2)⁻³ = 1/(−2)³ = 1/(−8) = −1/8.
3. Worked Examples — Laws Application
Example 1 — Simplify: (2⁻³ × 2⁵) ÷ 2² = 2⁻³⁺⁵⁻² = 2⁰ = 1.
Example 2 — Simplify: (x⁵ × x⁻²) / x³ = x⁵⁻²⁻³ = x⁰ = 1.
Example 3 — Simplify: (3⁻¹ + 2⁻¹)⁻¹. = (1/3 + 1/2)⁻¹ = (5/6)⁻¹ = 6/5.
Example 4 — Complex expression: [(2⁻¹ × 4⁻¹) ÷ 8⁻¹]⁻¹. First inside: 2⁻¹=1/2. 4⁻¹=1/4. Product = 1/8. Divide by 8⁻¹=1/8: (1/8)/(1/8) = 1. So [1]⁻¹ = 1.
Example 5 — Multiple laws: (a⁻²b³)⁻³ × (a⁴b⁻¹)². First bracket: a⁶b⁻⁹. Second: a⁸b⁻². Product: a¹⁴b⁻¹¹ = a¹⁴/b¹¹.
4. Scientific Notation (Standard Form)
Express very LARGE or very SMALL numbers as: a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
Large Numbers
150,000,000 km (Earth-Sun distance) = 1.5 × 10⁸ km. 384,000 km (Earth-Moon distance) = 3.84 × 10⁵ km. Avogadro's number: 6.022 × 10²³. Speed of light: 3 × 10⁸ m/s.
Small Numbers
0.000001 m (1 micron) = 1 × 10⁻⁶ m. 0.000000001 m (1 nm) = 1 × 10⁻⁹ m. Diameter of hydrogen atom: 1.06 × 10⁻¹⁰ m. Size of a virus: ~1 × 10⁻⁷ m.
Converting to Standard Form
Large number 45,600,000: Place decimal after first non-zero digit: 4.56. Count digits moved: 7 places left. = 4.56 × 10⁷.
Small number 0.0000032: Move decimal to right of first non-zero: 3.2. Count digits moved: 6 places right. = 3.2 × 10⁻⁶.
Comparing in Scientific Notation
Compare 5.2 × 10⁸ and 8.7 × 10⁷. 10⁸ > 10⁷, so 5.2×10⁸ is larger. When exponent differs, the larger exponent gives the larger number. When exponents are same, compare the coefficient a.
5. Common Mistakes
- Interpreting −aⁿ: −2⁴ = −(2⁴) = −16. (−2)⁴ = +16. Parentheses change the sign.
- a⁻ⁿ = negative number: NO — a⁻ⁿ = 1/aⁿ. 2⁻³ = 1/8, not −8.
- aᵐ × bⁿ ≠ (ab)ᵐ⁺ⁿ: Bases must be SAME to add exponents. 2³×3⁴ stays as is.
- Forgetting that a⁰ = 1 for ANY non-zero a: (1000)⁰ = 1, not 0.
6. AP Exam Focus
| Topic | Marks |
|---|---|
| Simplifying using laws | 3-4 |
| Negative exponents | 2-3 |
| Scientific notation (write/compare) | 2-3 |
| Word problems | 3-4 |
Key Exam Tips
- When simplifying, take it step by step. Apply ONE law at a time.
- For scientific notation: 'a' must be between 1 and 10. If the decimal goes beyond, adjust: 45×10⁶ = 4.5×10⁷.
- Negative exponents: if the final answer has negative exponents, rewrite with positive exponents in the denominator.
Quick Self-Test
- Simplify: (5⁻¹×2⁻¹) × 3⁻¹. (Answer: (1/5×1/2)×1/3 = 1/30.)
- Simplify: (a²b⁻³)² × (a⁻¹b)³. (Answer: a⁴b⁻⁶ × a⁻³b³ = a¹b⁻³ = a/b³.)
- Write 0.000000567 in standard form. (Answer: 5.67×10⁻⁷.)
- Value of 7⁰ + 2⁻¹? (Answer: 1 + 1/2 = 3/2.)
- Write 1.2×10⁻⁴ as ordinary decimal. (Answer: 0.00012.)
Advanced Worked Examples
Example — Complex Simplification: Simplify [{(625)^(−1/2)}^(−1/4)]². Inside: 625=5⁴. (625)^(−1/2) = (5⁴)^(−1/2) = 5^(−2) = 1/25. Then (1/25)^(−1/4) = 25^(1/4) = (5²)^(1/4) = 5^(1/2) = √5. Finally, [√5]² = 5.
Example — Comparing Quantities in Science: Which is larger: (a) size of a bacteria = 5×10⁻⁶ m or (b) size of a virus = 1.5×10⁻⁸ m? Compare exponents: 10⁻⁶ > 10⁻⁸, so bacteria is larger. How many times? (5×10⁻⁶)/(1.5×10⁻⁸) = (5/1.5)×10² = 3.33×100 ≈ 333 times larger.
Example — Real-world Magnitudes: Speed of light = 3×10⁸ m/s. One year = 3.156×10⁷ seconds. Distance light travels in one year (light-year) = (3×10⁸)×(3.156×10⁷) = 9.468×10¹⁵ m. 'A light-year is approximately 9.46×10¹² km — a UNIT of distance, not time. This is a common misconception.'
Example — Mass of Earth: Mass = 5.97×10²⁴ kg. Write this as an ordinary number. 5.97×10²⁴ = 5,970,000,000,000,000,000,000,000 kg. 'Writing even relatively simple scientific quantities requires scientific notation. Without it, astronomy and physics would be impractical.'
Understanding Scale — From the Universe to Atoms
| Object | Size in metres | Scientific Notation |
|---|---|---|
| Observable universe | 880,000,000,000,000,000,000,000,000 m | 8.8×10²⁶ m |
| Milky Way galaxy diameter | 1,000,000,000,000,000,000,000 m | 1×10²¹ m |
| Earth diameter | 12,742,000 m | 1.2742×10⁷ m |
| Human height | 1.7 m | 1.7×10⁰ m |
| Red blood cell | 0.000007 m | 7×10⁻⁶ m |
| Atom (hydrogen) | 0.0000000001 m | 1×10⁻¹⁰ m |
| Proton | 0.000000000000001 m | 1×10⁻¹⁵ m |
Laws of Exponents — Combined Application
Simplify: (a⁻²b)³ × (ab²)⁻² ÷ (a³b⁻¹)². First term: a⁻⁶b³. Second: a⁻²b⁻⁴. Product: a⁻⁸b⁻¹. Divide by third: a⁻⁸b⁻¹/a⁶b⁻² = a⁻¹⁴b¹ = b/a¹⁴. Write with positive exponents.
Exponential Growth vs Linear Growth
Linear growth: 2, 4, 6, 8, 10 (adds 2 each time). Exponential growth: 2, 4, 8, 16, 32 (multiplies by 2 each time). 'Exponential growth starts slow but quickly OUTPACES linear growth. This is why compound interest beats simple interest over long periods — and why pandemics spread so rapidly.'
AP Specific — ISRO and Scientific Notation
ISRO's Chandrayaan-3 traveled approximately 384,400 km (3.844×10⁵ km) to the Moon. Sriharikota launch site (SHAR) is at approximately 13.72°N, 80.23°E. The PSLV rocket's payload capacity to LEO (Low Earth Orbit) is about 1,750 kg = 1.75×10³ kg. GSLV MkIII can lift 4,000 kg = 4×10³ kg to GTO.
