🎯 What You're About to Master
Index laws are the shortcuts that let you manipulate powers without writing them out. Instead of expanding x³ × x⁵ as (x·x·x) × (x·x·x·x·x), you just add the powers to get x⁸. This chapter teaches you how to apply these rules with negative powers, fractional powers, and unknowns hidden in exponents. Master this, and you'll solve complex equations that look scary but follow simple patterns.
Why this matters
Index laws are everywhere in higher maths, physics, and even computer science. They're also high-frequency exam material—questions on this topic almost always combine multiple laws in one problem, which is why understanding why each law works is crucial.
📐 The Eight Core Index Laws
Law 1 & 2: The Trivial Rules
a¹ = a and a⁰ = 1
Any number to the power of 1 is itself. Any non-zero number to the power of 0 is always 1.
Why? Think of the power as "how many times you multiply the base by itself."
- a¹: Multiply once → just a
- a⁰: Multiply zero times → you get the "multiplicative identity," which is 1
Law 3: Multiply — Add Powers
aᵐ × aⁿ = aᵐ⁺ⁿ
When you multiply powers with the same base, add the exponents.
Example: x³ × x⁵ = (x·x·x) × (x·x·x·x·x) = x⁸ = x⁽³⁺⁵⁾
Why? You're counting how many x's appear in total. 3 of them, then 5 more = 8 total.
Law 4: Divide — Subtract Powers
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
When you divide powers with the same base, subtract the exponents.
Example: x⁷ ÷ x² = (x·x·x·x·x·x·x) / (x·x) = x⁵ = x⁽⁷⁻²⁾
Why? You're cancelling matching x's from top and bottom. Start with 7, remove 2 → 5 left.
Law 5: Power of a Power — Multiply Powers
(aᵐ)ⁿ = aᵐⁿ
When you raise a power to another power, multiply the exponents.
Example: (x³)⁴ = (x·x·x) raised to the 4th power = (x·x·x) × (x·x·x) × (x·x·x) × (x·x·x) = x¹² = x⁽³×⁴⁾
Why? You're repeating the group. If one group has 3 x's and you repeat it 4 times, you have 3 × 4 = 12 x's total.
Law 6: Power of a Product
(ab)ⁿ = aⁿbⁿ
Raise each factor in the product to the power separately, then multiply.
Example: (2x)³ = 2³ × x³ = 8x³
Why? (2x)³ = (2x) × (2x) × (2x) = (2 × 2 × 2) × (x × x × x) = 8x³. You can rearrange multiplication.
Law 7: Power of a Fraction
(a/b)ⁿ = aⁿ/bⁿ
Raise the numerator and denominator to the power separately.
Example: (x/2)² = x²/4
Why? (x/2)² = (x/2) × (x/2) = (x × x) / (2 × 2) = x²/4
Law 8: Negative Powers are Reciprocals
a⁻ⁿ = 1/aⁿ
A negative power flips the fraction. It's the reciprocal.
Examples:
- x⁻¹ = 1/x
- x⁻³ = 1/x³
- (2/3)⁻² = (3/2)² = 9/4
Why? Using the divide rule: x⁰ ÷ x³ = x⁻³. But x⁰ = 1, so 1 ÷ x³ = 1/x³.
1
Simplify: (3x²) × (2x⁵)
➗ Fractional & Negative Fractional Powers
Understanding Fractional Powers as Roots
a^(1/n) = ⁿ√a
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