Laws of Indices
Every index manipulation rests on five laws: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(m×n), (ab)^n = a^n·b^n, and a^0 = 1 with a^(−n) = 1/a^n. The CAT skill is not memorising them but choosing the right one fast and forcing every term onto a common base. To solve 2^x = 32, write 32 as 2^5 and equate exponents; to compare 2^40 and 3^30, rewrite as (2^4)^10 = 16^10 versus (3^3)^10 = 27^10, so the larger base wins. When an equation has the unknown in the exponent, matching bases beats logarithms almost every time. Watch order of operations: a^m^n means a^(m^n), not (a^m)^n — exponentiation is right-associative, a favourite CAT trap.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Laws of indices
| Product of same base | a^m × a^n = a^(m+n) |
|---|---|
| Quotient of same base | a^m ÷ a^n = a^(m−n) |
| Power of a power | (a^m)^n = a^(m×n) |
| Power of a product / quotient | (ab)^n = a^n b^n; (a/b)^n = a^n / b^n |
| Zero and negative index | a^0 = 1; a^(−n) = 1 / a^n |
Surds & rationalization
| Fractional index = root | a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m) |
|---|---|
| Product / quotient of roots | ⁿ√a × ⁿ√b = ⁿ√(ab); ⁿ√a ÷ ⁿ√b = ⁿ√(a/b) |
| Conjugate of (√a + √b) | √a − √b (their product = a − b) |
| Rationalize 1/(√a + √b) | (√a − √b) / (a − b) |
| Compound surd √(a ± 2√b) | √x ± √y where x + y = a, xy = b |