Indices/roots; surds (extended)

Indices (or powers/exponents) represent repeated multiplication. The expression a^n means 'a multiplied by itself n times', where a is the base and n is the index or power.

Key Index Laws:

1. Multiplication: a^m × a^n = a^(m+n)
2. Division: a^m ÷ a^n = a^(m−n)
3. Power of a power: (a^m)^n = a^(mn)
4. Power of a product: (ab)^n = a^n × b^n
5. Zero index: a^0 = 1 (where a ≠ 0)
6. Negative index: a^(−n) = 1/a^n
7. Fractional index: a^(1/n) = ⁿ√a (the nth root of a)
8. Combined fractional: a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m)

Surds are irrational numbers expressed as roots that cannot be simplified to whole numbers (e.g., √2, √5, ³√7). They provide exact values rather than decimal approximations.

Key Surd Rules:

1. Multiplication: √a × √b = √(ab)
2. Division: √a ÷ √b = √(a/b)
3. Simplification: √(a²b) = a√b
4. Rationalising the denominator: Remove surds from denominators by multiplying by a suitable form

Memory Aid (BEDMAS for indices): 'Big Elephants Don't Make Amazing Singers' — Base, Exponent, Division (subtract), Multiplication (add), Another power (multiply), Subtract (for division)

Understanding these laws is fundamental for Extended IGCSE, as they appear in algebra, equations, and scientific notation questions worth 3–6 marks.

Indices in Real Life:

Exponential growth models use indices constantly. When bacteria double every hour, a population of 100 becomes 100 × 2^t after t hours. After 5 hours: 100 × 2^5 = 3,200 bacteria. Financial compound interest follows A = P(1 + r)^n, where indices calculate investment growth over n years.

Computer storage uses powers of 2: 1 kilobyte = 2^10 bytes (1,024), 1 megabyte = 2^20 bytes. Understanding indices helps calculate storage capacity efficiently.

Negative indices model decay: radioactive half-life uses formulas like N = N₀(1/2)^t = N₀ × 2^(−t). If a substance has half-life 3 years, after 9 years only (1/2)^3 = 1/8 remains.

Surds in Architecture and Design:

The diagonal of a square with side length 1 is exactly √2 ≈ 1.414... This irrational ratio appears in A4 paper (aspect ratio 1:√2), ensuring proportions remain constant when folding. Ancient Greeks discovered √2 when measuring diagonals, revolutionizing mathematics.

Engineering and Physics require exact calculations. The distance formula d = √((x₂−x₁)² + (y₂−y₁)²) uses surds for precision. When calculating forces, engineers use √3 in three-phase electrical systems rather than 1.732, maintaining accuracy through calculations.

Analogy: Think of indices as mathematical shortcuts — instead of writing 2×2×2×2×2, write 2^5. Surds are like exact GPS coordinates versus approximate addresses; √50 is precisely √50, while 7.071... loses information.

Mastering these concepts enables solving complex algebraic equations and understanding exponential relationships critical in science and economics.

Example 1: Simplify (27^(2/3)) × (9^(−1/2))

Solution:

Step 1: Express as prime bases: 27 = 3³, 9 = 3²

Step 2: Apply fractional index law: 27^(2/3) = (3³)^(2/3) = 3^(3×2/3) = 3² = 9

Step 3: Simplify second term: 9^(−1/2) = (3²)^(−1/2) = 3^(2×−1/2) = 3^(−1) = 1/3

Step 4: Multiply: 9 × 1/3 = 3

Examiner note: Show each step clearly. Converting to prime bases earns method marks even if final answer is incorrect.

Example 2: Simplify (3√8 − √50 + √32) and express in form a√2

Solution:

Step 1: Simplify each surd: √8 = √(4×2) = 2√2

Step 2: √50 = √(25×2) = 5√2

Step 3: √32 = √(16×2) = 4√2

Step 4: Substitute: 3(2√2) − 5√2 + 4√2 = 6√2 − 5√2 + 4√2 = 5√2

Examiner note: Always look for perfect square factors (4, 9, 16, 25...). This question awards 3 marks: 1 for each correct simplification, 1 for final combination.

Example 3: Rationalize and simplify: (6)/(3−√5)

Solution:

Step 1: Multiply by conjugate: (6/(3−√5)) × ((3+√5)/(3+√5))

Step 2: Expand numerator: 6(3+√5) = 18 + 6√5

Step 3: Expand denominator: (3−√5)(3+√5) = 9 − 5 = 4

Step 4: Simplify: (18 + 6√5)/4 = 9/2 + (3√5)/2

Examiner note: Difference of two squares eliminates the surd. Show all working for full marks.