Exponents and Powers — A Shorthand for Repeated Multiplication
"Writing 10,000,000 is tedious. Writing 10⁷ is elegant. Exponents are SHORTHAND for repeated multiplication."
1. What Is an Exponent?
Exponent (or power/index) tells us how many times a number (the BASE) is multiplied by itself.
Expression: aⁿ = a × a × a × ... (n times).
- a = base (the number being multiplied)
- n = exponent/power (how many times)
Read as: 'a raised to the power n' or 'a to the n-th power.'
| Expression | Meaning | Value |
|---|---|---|
| 2⁵ | 2 × 2 × 2 × 2 × 2 | 32 |
| 5³ | 5 × 5 × 5 | 125 |
| (−3)⁴ | (−3) × (−3) × (−3) × (−3) | 81 |
| (−2)³ | (−2) × (−2) × (−2) | −8 |
| 10² | 10 × 10 | 100 |
| 10⁶ | 10 × 10 × 10 × 10 × 10 × 10 | 1,000,000 |
Important Distinctions
'(−3)⁴ means (−3) × (−3) × (−3) × (−3) = +81. But −3⁴ means −(3⁴) = −81.' Pay attention to parentheses!
| Expression | Reading | Result |
|---|---|---|
| (−2)³ | 'Cube of negative 2' | −8 |
| −2³ | 'Negative of 2 cubed' | −8 (same here) |
| (−2)⁴ | 'Fourth power of negative 2' | +16 |
| −2⁴ | 'Negative of 2 to the fourth' | −16 |
2. Laws of Exponents
Law 1: Product of Powers (Same Base)
aᵐ × aⁿ = aᵐ⁺ⁿ
'When multiplying powers with the SAME base, ADD the exponents.'
| Example | Application | Result |
|---|---|---|
| 2³ × 2⁵ | 2³⁺⁵ | 2⁸ = 256 |
| 5² × 5⁶ | 5²⁺⁶ | 5⁸ = 390,625 |
| x⁴ × x³ | x⁴⁺³ | x⁷ |
| a² × a | a²⁺¹ | a³ |
Law 2: Quotient of Powers (Same Base)
aᵐ ÷ aⁿ = aᵐ⁻ⁿ (where m > n)
'When dividing powers with the SAME base, SUBTRACT the exponents.'
| Example | Application | Result |
|---|---|---|
| 2⁸ ÷ 2³ | 2⁸⁻³ | 2⁵ = 32 |
| 7⁶ ÷ 7² | 7⁶⁻² | 7⁴ = 2401 |
| x⁷ ÷ x⁴ | x⁷⁻⁴ | x³ |
Law 3: Power of a Power
(aᵐ)ⁿ = aᵐⁿ
'When raising a power to another power, MULTIPLY the exponents.'
| Example | Application | Result |
|---|---|---|
| (2³)² | 2³ˣ² | 2⁶ = 64 |
| (5²)⁴ | 5²ˣ⁴ | 5⁸ = 390,625 |
| (x²)⁵ | x²ˣ⁵ | x¹⁰ |
Law 4: Power of a Product
(a × b)ᵐ = aᵐ × bᵐ
'Raise EACH factor to the power.'
| Example | Application | Result |
|---|---|---|
| (2 × 3)² | 2² × 3² | 4 × 9 = 36 |
| (2x)³ | 2³ × x³ | 8x³ |
| (−2a)⁴ | (−2)⁴ × a⁴ | 16a⁴ |
Law 5: Power of a Quotient
(a ÷ b)ᵐ = aᵐ ÷ bᵐ (b ≠ 0)
'Raise BOTH numerator and denominator to the power.'
| Example | Application | Result |
|---|---|---|
| (2/3)² | 2²/3² | 4/9 |
| (x/5)³ | x³/5³ | x³/125 |
3. Zero Exponent
Any non-zero number raised to the power ZERO equals 1. a⁰ = 1 (a ≠ 0)
| Expression | Value | Reason |
|---|---|---|
| 5⁰ | 1 | By definition |
| (−3)⁰ | 1 | Any non-zero number |
| 1000⁰ | 1 | Any non-zero number |
| 0⁰ | UNDEFINED | Convention |
'Proof using quotient law: aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰. But aᵐ ÷ aᵐ = 1. So a⁰ = 1.'
4. Negative Exponents
a⁻ⁿ = 1/aⁿ (a ≠ 0)
'A negative exponent means RECIPROCAL. It does NOT make the number negative.'
| Expression | Meaning | Value |
|---|---|---|
| 2⁻³ | 1/2³ | 1/8 = 0.125 |
| 10⁻² | 1/10² | 1/100 = 0.01 |
| (−3)⁻² | 1/(−3)² | 1/9 |
| x⁻⁵ | 1/x⁵ | — |
Also: 1/a⁻ⁿ = aⁿ.
Example: Simplify 3⁻² × 3⁻³. 3⁻²⁻³ = 3⁻⁵ = 1/3⁵ = 1/243.
5. Expressing Large Numbers in Standard Form (Scientific Notation)
Standard form: A × 10ⁿ where 1 ≤ A < 10 and n is an integer (positive or negative).
'This is how scientists represent VERY LARGE and VERY SMALL numbers compactly.'
Converting Large Numbers to Standard Form
'Move the decimal point to the LEFT until only ONE non-zero digit is to the left. Count the number of places moved — that becomes the positive exponent of 10.'
| Number | Steps | Standard Form |
|---|---|---|
| 3,000,000 | 3.0 × 10⁶ (move 6 left) | 3 × 10⁶ |
| 5,72,00,00,000 (India) | 5.72 × 10⁹ (move 9 left) | 5.72 × 10⁹ |
| 150,000,000 (Sun-Earth distance in km) | 1.5 × 10⁸ | 1.5 × 10⁸ |
| 8,50,00,00,000 | 8.5 × 10⁹ | 8.5 × 10⁹ |
AP Context
'AP population (approx): 5.4 crore = 54,000,000 = 5.4 × 10⁷.' 'Light travels at 300,000,000 m/s = 3 × 10⁸ m/s.' 'Mass of Earth: 5,970,000,000,000,000,000,000,000 kg = 5.97 × 10²⁴ kg.'
Converting Standard Form Back
'Move the decimal point to the RIGHT by the exponent value. Add zeros as needed.'
| Standard Form | Process | Ordinary Number |
|---|---|---|
| 2.5 × 10⁴ | 2.5 → 25000 | 25,000 |
| 1.2 × 10⁶ | 1.2 → 1200000 | 1,200,000 |
6. Comparing Powers
Method: Express both numbers using the SAME exponent or express both in standard form.
Example: Which is larger: 2⁸ or 3⁵? 2⁸ = 256, 3⁵ = 243. 2⁸ > 3⁵.
Example: Compare 4.5 × 10⁷ and 7.2 × 10⁶. 4.5 × 10⁷ = 45 × 10⁶. Since 45 > 7.2, 4.5 × 10⁷ > 7.2 × 10⁶.
7. Common Mistakes and Fixes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| 2³ = 6 | Confusing exponent with multiplication | 2³ = 2 × 2 × 2 = 8 |
| a² × a³ = a⁶ | Adding exponents wrong | a²⁺³ = a⁵ |
| (a²)³ = a⁵ | Multiplying when should multiply | (a²)³ = a²ˣ³ = a⁶ |
| (−2)⁴ = −16 | Odd number of negatives? No — even exponent | (−2)⁴ = +16 |
| 2⁻³ = −8 | Negative exponent ≠ negative result | 2⁻³ = 1/2³ = 1/8 |
| 0⁰ = 1 | Ambiguous and undefined by convention | 0⁰ is NOT defined |
| a³b² = (ab)⁵ | Cannot combine different bases | a³b² stays as is |
8. AP SSC Exam Focus
| Topic | Marks | Question Type |
|---|---|---|
| Laws of exponents — multiplication | 2-3 | Simplify expressions |
| Laws of exponents — division | 2-3 | Simplify expressions |
| Zero and negative exponents | 2-3 | Evaluate |
| Standard form / scientific notation | 3-4 | Convert to and from |
| Combined application | 3-4 | Multi-step simplification |
Quick Self-Test
Q1. Simplify: 2⁵ × 2³. A1. 2⁵⁺³ = 2⁸ = 256.
Q2. Simplify: 7⁶ ÷ 7². A2. 7⁶⁻² = 7⁴ = 2401.
Q3. Simplify: (x³)² × x⁴. A3. x⁶ × x⁴ = x¹⁰.
Q4. Evaluate: 5⁻². A4. 1/25 = 0.04.
Q5. Express 0.000000345 in standard form. A5. 3.45 × 10⁻⁷.
Q6. Express 6.02 × 10²³ as an ordinary number. A6. 602,000,000,000,000,000,000,000 (602 sextillion / 6.02 × 10²³).
Q7. Simplify: (2a²b)³. A7. 2³ × (a²)³ × b³ = 8a⁶b³.
Q8. Is 2⁵⁰ > 3³⁰? (Hint: take log or compare approximate powers) A8. 2⁵⁰ = (2⁵)¹⁰ = 32¹⁰. 3³⁰ = (3³)¹⁰ = 27¹⁰. Since 32 > 27, 2⁵⁰ > 3³⁰.
