IB Math AI HL — Non-integer exponents (HL only)

Key Idea: Standard form writes any number as a × 10ⁿ — where 1 ≤ a < 10 and n is an integer. Used for very large numbers (distance to the Sun ≈ 1.5 × 10¹¹ m) and very small numbers (size of an atom ≈ 1 × 10⁻¹⁰ m).

The structure: a × 10ⁿ

a — the coefficient

Always 1 ≤ a < 10. Holds the significant figures. Example: 47,000 = 4.7 × 10⁴ (a = 4.7 ✓)

n — the exponent

Any integer. Counts how far the decimal moved. n > 0 → large number (≥ 10). n < 0 → small number (< 1).

Number → standard form (move decimal until a is between 1 and 10): 47,000 → decimal moves 4 left → 4.7 × 10⁴ 0.0038 → decimal moves 3 right → 3.8 × 10⁻³ Standard form → ordinary number (reverse it): 4.7 × 10⁴ → move decimal 4 right → 47,000 3.8 × 10⁻³ → move decimal 3 left → 0.0038

Tip: Negative exponent ≠ negative number. 3.2 × 10⁻⁴ = 0.00032 — positive, just very small. Recheck a after every calculation. 24.6 × 10³ is not valid — adjust to 2.46 × 10⁴. GDC shows 4.7E4 — not a valid written answer. Always write 4.7 × 10⁴.

🔢 Calculations in standard form

Standard form rarely appears alone. It shows up as the last part of a logs, growth, or finance question — "write your answer in the form a × 10ᵏ". Miss the format, lose the mark.