What this chapter is about
This chapter explores the world of exponents and powers, building on what you learnt in Class 7. You will work with powers that have negative bases, compare very large and very small numbers, and learn the laws that make calculations with powers much easier.
Powers are a shorthand for repeated multiplication. Instead of writing 2 × 2 × 2 × 2 × 2, we write 2⁵. This notation becomes essential when dealing with extremely large numbers like the distance to stars in metres or extremely small numbers like the size of bacteria. Scientists, engineers and economists use powers daily.
By the end of this chapter, you should be able to apply laws of exponents confidently, express numbers in standard form, and simplify expressions involving powers with the same base or the same exponent.
Key ideas
- Power notation: In aⁿ, the number a is called the base and n is called the exponent (or index). It means a multiplied by itself n times.
- Laws of exponents: When multiplying powers with the same base, add the exponents (aᵐ × aⁿ = aᵐ⁺ⁿ). When dividing, subtract them (aᵐ ÷ aⁿ = aᵐ⁻ⁿ). When raising a power to another power, multiply the exponents ((aᵐ)ⁿ = aᵐⁿ).
- Power of a product or quotient: (a × b)ⁿ = aⁿ × bⁿ and (a/b)ⁿ = aⁿ/bⁿ.
- Zero exponent: Any non-zero number raised to the power 0 equals 1 (a⁰ = 1, where a ≠ 0).
- Negative exponents: a⁻ⁿ means 1/aⁿ. It flips the base into the denominator.
- Negative bases: When a negative number is raised to an even power, the result is positive; when raised to an odd power, the result is negative.
- Standard form: Very large or small numbers are written as k × 10ⁿ, where 1 ≤ k < 10 and n is an integer.
Formulas and facts to remember
- Rule: Product of powers · Formula: aᵐ × aⁿ = aᵐ⁺ⁿ · Meaning: Same base: add exponents
- Rule: Quotient of powers · Formula: aᵐ ÷ aⁿ = aᵐ⁻ⁿ · Meaning: Same base: subtract exponents
- Rule: Power of a power · Formula: (aᵐ)ⁿ = aᵐⁿ · Meaning: Multiply the exponents
- Rule: Power of a product · Formula: (ab)ⁿ = aⁿbⁿ · Meaning: Distribute the exponent
- Rule: Power of a quotient · Formula: (a/b)ⁿ = aⁿ/bⁿ · Meaning: Distribute the exponent
- Rule: Zero exponent · Formula: a⁰ = 1 (a ≠ 0) · Meaning: Any non-zero base to power 0 is 1
- Rule: Negative exponent · Formula: a⁻ⁿ = 1/aⁿ · Meaning: Flip to denominator, make exponent positive
Worked examples
### Example 1: Simplify 3⁴ × 3² ÷ 3⁵
Step 1: Use the product rule for 3⁴ × 3² 3⁴ × 3² = 3⁴⁺² = 3⁶
Step 2: Now divide by 3⁵ using the quotient rule 3⁶ ÷ 3⁵ = 3⁶⁻⁵ = 3¹ = 3
Answer: 3
### Example 2: Write 0.000045 in standard form
Step 1: Move the decimal point to get a number between 1 and 10 0.000045 → 4.5 (decimal moved 5 places to the right)
Step 2: Since we moved right, the exponent of 10 is negative 0.000045 = 4.5 × 10⁻⁵
Answer: 4.5 × 10⁻⁵
### Example 3: Evaluate (−2)⁴ and (−2)⁵
For (−2)⁴: (−2) × (−2) × (−2) × (−2) = 4 × 4 = 16 Even power of a negative base gives a positive result.
For (−2)⁵: (−2)⁴ × (−2) = 16 × (−2) = −32 Odd power of a negative base gives a negative result.
Answers: (−2)⁴ = 16 and (−2)⁵ = −32
Common mistakes
- Writing 2³ × 2⁴ = 2¹² (multiplying exponents instead of adding) → When bases are same and you multiply, add the exponents: 2³ × 2⁴ = 2⁷.
- Thinking (−3)² and −3² are the same → (−3)² = 9 but −3² = −9, because without brackets only 3 is squared, then the minus sign is applied.
- Believing a⁰ = 0 → Any non-zero number to the power 0 equals 1, not 0.
- Forgetting to flip when exponent is negative → 5⁻² means 1/5² = 1/25, not −25.
- Confusing (2³)² with 2³ × 2² → (2³)² = 2⁶ (multiply exponents), while 2³ × 2² = 2⁵ (add exponents).
Quick revision
- aᵐ × aⁿ = aᵐ⁺ⁿ (add exponents when multiplying same bases).
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ (subtract exponents when dividing same bases).
- a⁻ⁿ = 1/aⁿ (negative exponent means reciprocal).
- a⁰ = 1 for any a ≠ 0.
- Standard form: k × 10ⁿ where 1 ≤ k < 10.
- Negative base with even power → positive; with odd power → negative.