A negative power does not make a number negative — it flips it into a fraction. And most surds are hiding a perfect square factor, waiting to be simplified.
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Every law of indices comes from the same idea — multiplying repeats an addition of powers, dividing repeats a subtraction.
📊 The laws of indices
| Law | Rule | Example |
|---|---|---|
| Multiplying | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
| Dividing | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 2⁷ ÷ 2³ = 2⁴ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ |
| Zero power | a⁰ = 1 (for any a ≠ 0) | 5⁰ = 1 |
| Negative power | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
Fractional powers are roots: a^(1/n) means the nth root of a. So 27^(1/3) = 3 (the cube root), and 8^(2/3) means (cube root of 8)² = 2² = 4.
A surd is an irrational root that cannot be simplified to a whole number — but it can often be simplified to a SMALLER surd.
📌 Simplifying a surd
Find the largest perfect-square factor of the number under the root.
Split it: √50 = √(25×2) = √25 × √2 = 5√2.
Combining surds: only surds with the SAME number under the root can be added or subtracted directly, like ordinary terms — 3√2 + 5√2 = 8√2.
Rationalising the denominator: a surd should never be left in the denominator. 1/√2 = (1×√2)/(√2×√2) = √2/2.
Standard form writes very large or very small numbers using powers of 10 — exactly what scientific notation is built for.
✨ Standard form examples
Large number
45,000,000
4.5 × 10⁷
Small number
0.00032
3.2 × 10⁻⁴
Pro tip: the power of 10 equals how many places the decimal point moves. Moving it LEFT (for a large number) gives a POSITIVE power; moving it RIGHT (for a small number) gives a NEGATIVE power.
12 questions on the laws of indices and simplifying surds.
