algebra
Algebraic Roots and indices
Algebraic Roots and Indices
GCSE Mathematics📐 What are Roots and Indices?
Indices (or exponents) show how many times a number is multiplied by itself. Roots are the inverse operation - finding which number multiplied by itself gives the original number. Together, they form the foundation of powers and surds in algebra.
Index (Power): 5³ = 5 × 5 × 5 = 125
Root: √25 = 5 (because 5 × 5 = 25)
Algebraic: x⁴ × x³ = x⁷
Root: √25 = 5 (because 5 × 5 = 25)
Algebraic: x⁴ × x³ = x⁷
📏 Laws of Indices
✖️ Law 1: Multiplication (Add indices)
aᵐ × aⁿ = aᵐ⁺ⁿ
Example 1
x³ × x⁴ = x³⁺⁴ = x⁷
Example 2
y² × y⁵ = y²⁺⁵ = y⁷
Example 3
3x² × 2x⁴ = (3×2)x²⁺⁴ = 6x⁶
➗ Law 2: Division (Subtract indices)
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Example 1
x⁷ ÷ x³ = x⁷⁻³ = x⁴
Example 2
y⁶ ÷ y² = y⁶⁻² = y⁴
Example 3
12x⁵ ÷ 3x² = (12÷3)x⁵⁻² = 4x³
⬆️ Law 3: Power of a Power (Multiply indices)
(aᵐ)ⁿ = aᵐ×ⁿ
Example 1
(x³)² = x³×² = x⁶
Example 2
(y⁴)³ = y⁴׳ = y¹²
Example 3
(2x²)³ = 2³ × (x²)³ = 8x⁶
➖ Law 4: Negative Indices
a⁻ⁿ = 1/aⁿ
Example 1
x⁻³ = 1/x³
Example 2
2y⁻⁴ = 2/y⁴
Example 3
(2x)⁻² = 1/(2x)² = 1/(4x²)
0️⃣ Law 5: Zero Index
a⁰ = 1 (where a ≠ 0)
Example 1
x⁰ = 1
Example 2
5⁰ = 1
Example 3
(2x³)⁰ = 1
√ Roots and Surds
🔲 Square Roots
√a × √a = a
√16 = 4 (because 4² = 16)
√x⁶ = x³ (because (x³)² = x⁶)
📦 Cube Roots
∛a × ∛a × ∛a = a
∛27 = 3 (because 3³ = 27)
∛x⁹ = x³ (because (x³)³ = x⁹)
🔄 Fractional Indices
a^(1/n) = ⁿ√a
x^(1/2) = √x
x^(1/3) = ∛x
🧮 Interactive Indices Calculator
Enter expressions and see how indices laws work!
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✏️ Practice Questions
Simplify: x³ × x⁴
Score: 0
Attempted: 0
🌟 Challenge Problems
Simplify: (2x³y²)⁴
Simplify: (8x⁶)^(1/3)
Simplify: (4x⁴)^(1/2) × x³
⚠️ Common Mistakes
❌ Wrong: Adding when multiplying powers
x³ × x⁴ = x¹² ❌
Correct: x³ × x⁴ = x⁷ (add indices)
✅ Correct: Remember the laws
Multiplication → Add indices
Division → Subtract indices
Power of power → Multiply indices
❌ Wrong: Forgetting about coefficients
2x³ × 3x⁴ = 5x⁷ ❌
Correct: 2x³ × 3x⁴ = 6x⁷
⚠️ Negative indices are reciprocals
x⁻³ = 1/x³, not -x³
🌍 Where Indices Are Used
💰 Compound Interest
A = P(1 + r)ⁿ
Where n is the number of years (index)
📏 Scientific Notation
3.2 × 10⁶ = 3,200,000
Powers of 10 make big numbers manageable
📊 Area and Volume
Area = length² (square units)
Volume = length³ (cubic units)
🧬 Exponential Growth
Bacteria growth: N = N₀ × 2ᵗ
Population doubles each hour