Indices &
Standard Form

Law: When multiplying powers with the same base, you add the exponents: a^m × aⁿ = a^m+n. The base stays the same, only the exponents change.

Example: x³ × x⁵ = x³⁺⁵ = x⁸ or 2⁴ × 2² = 2⁴⁺² = 2⁶ = 64

💰 Real life: this is how billionaires' wealth compounds. Growing by a fixed percentage every year is the same maths as base^x above — each year's total gets multiplied by the same number again, so the curve stays flat for a while, then rockets upward. This is called compound growth (the same rule behind compound interest on a bank loan or savings account).

In 2025, the collective wealth of US billionaires grew by about 22% — from roughly $6.7 trillion to $8.2 trillion in a single year. That's a "base" of 1.22: multiply by 1.22, year after year, and the numbers explode just like the steep end of the curves above — source: The Guardian, Nov 2025.

The formula is: A = P × (1 + r/100)^x, where P is the starting amount, r is the percentage growth rate, and x is the number of years — the exponent x does exactly the same job here as it does in the questions below. Growing at 22% a year means wealth roughly doubles every ~3.5 years, because 1.22^3.5 ≈ 2.

Law: When dividing powers with the same base, you subtract the exponents: a^m ÷ aⁿ = a^m-n. The base stays the same.

Example: x⁷ ÷ x² = x^7-2 = x⁵ or 5⁶ ÷ 5² = 5^6-2 = 5⁴ = 625

Law: When raising a power to another power, you multiply the exponents: (a^m)ⁿ = a^m×n. Think of it as "power of a power, multiply the exponent by the power."

Example: (x³)⁴ = x^3×4 = x¹² or (2⁵)² = 2^5×2 = 2¹⁰ = 1024

Zero exponent: Any non-zero number to the power of 0 is 1: a⁰ = 1 (where a ≠ 0).

Negative exponent: A negative exponent means the reciprocal (one divided by): a^-n = 1/(aⁿ). Example: 2^-3 = 1/2³ = 1/8

Combining laws: Harder questions mix multiplying, dividing and power-of-a-power in one expression. Work left to right, and if there's a fraction, simplify the numerator and denominator separately before combining them.

Example: (8² × 8⁵) ÷ 8³ = 8⁷ ÷ 8³ = 8⁴ — simplify inside the brackets first, then apply the next law.

Definition: Standard form writes a number as a × 10^b, where a is between 1 and 9.9̇ (it can be negative) and b is a whole number (also can be negative). It's used to write very large or very small numbers concisely, and to compare them easily.

Example: 3 × 10⁸ = 300,000,000. The "3" gives the digits used; the "10⁸" tells us the scale — how many places the first digit sits from the units column.

Method: Keep dividing the number by 10 until you get a number between 1 and 9.9̇ — that's a. Count how many times you divided (how many places the decimal point moved left) — that's b.

Example: 4000 → 4 × 10³ (the decimal point moved 3 places left).

Method: For numbers smaller than 1, the decimal point moves right to reach the first non-zero digit — that makes b negative. A quick trick: for negative powers, the power matches the number of leading zeroes (including the one before the decimal point).

Example: 0.002 → 2 × 10^-3 (the point moved 3 places right).

Method: This is the reverse process. The power of 10 tells you how many places to move the decimal point — right for a positive power (number gets bigger), left for a negative power (number gets smaller). Pad with zeroes as needed.

Example: 9 × 10⁴ = 90,000 and 8.7 × 10^-3 = 0.0087

Method: Multiply/divide the front numbers together, and use the laws of indices on the powers of 10 (add for ×, subtract for ÷). If the front number ends up outside 1–9.9̇, renormalise it and adjust the power of 10 to compensate.

Example: (7 × 10³) × (6 × 10¹⁰) = 42 × 10¹³ = 4.2 × 10¹⁴ — 42 isn't in range, so it becomes 4.2 and the power increases by 1 to compensate.