An index is a count of how many times
2⁵ means 2 multiplied by itself five times. The 2 is the base and the 5 is the index, power or exponent — three words for the same thing, all of which appear in exam questions.
Every law below follows from that definition, which is why they are worth deriving once rather than memorising six separate rules. 2³ × 2⁴ is three 2s multiplied by four 2s, which is seven 2s — so the indices add.
Why a⁰ = 1 and why a negative index means a reciprocal
Follow the division law downwards: 2³ ÷ 2³ = 2⁰, and anything divided by itself is 1 — so 2⁰ = 1. Continue: 2³ ÷ 2⁵ = 2⁻², and cancelling gives 1/2² — so a negative index means one over. Neither is an arbitrary convention; both are forced by the law that was already there.
Fractional indices are roots
What could 9^(1/2) mean? Whatever it is, the multiplication law says 9^(1/2) × 9^(1/2) = 9¹ = 9. So it is the number that multiplies by itself to give 9 — the square root, 3.
The same argument gives a^(1/3) as the cube root, and generally a^(1/n) = ⁿ√a. For a general fraction, take the root first and then the power: 8^(2/3) means the cube root of 8, which is 2, then squared, giving 4. Rooting first keeps the numbers small.
Evaluate 16^(3/4), 5⁻² and (2⁻³)⁻² without a calculator.
16^(3/4): the denominator 4 is the root, so take the fourth root of 16 first, which is 2.Rooting first keeps the arithmetic small — 16³ would be 4096 before you even started.- Then raise to the numerator:
2³ = 8. 5⁻² = 1/5² = 1/25.The minus sign means reciprocal, not a negative answer. 5⁻² is not −25.(2⁻³)⁻² = 2^(−3 × −2) = 2⁶ = 64.The power law multiplies the indices, and two negatives give a positive.
16^(3/4) = 8; 5⁻² = 1/25; (2⁻³)⁻² = 64
A negative index never gives a negative answer
3⁻² = 1/9, which is positive and small. Students write −9 remarkably often. The index sign controls whether you go up or down — it has nothing to do with the sign of the result.
Standard form
The distance to the sun is about 150 000 000 000 metres, and the diameter of an atom about 0.0000000001 m. Both are unreadable, and counting the zeros is where the mistakes happen.
Standard form writes every number as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. Those two conditions are the whole definition, and both are marked: 15 × 10¹⁰ is arithmetically correct but not in standard form, because 15 is not below 10.
Work out (3.2 × 10⁵) × (4 × 10⁻³), giving the answer in standard form.
- Multiply the number parts:
3.2 × 4 = 12.8.Handle the digits and the powers separately — that is what standard form is for. - Add the indices:
10⁵ × 10⁻³ = 10².The multiplication law, with a negative index simply subtracting. - So far
12.8 × 10²— correct, but not standard form.12.8 is not between 1 and 10, so one more step is needed. - Adjust:
12.8 = 1.28 × 10¹, so the answer is1.28 × 10³.Moving the point one place left increases the index by one. Check: 1280 ✓
1.28 × 10³
Powers, roots and the calculator
The syllabus expects square and cube numbers up to reasonable limits to be known by sight, because recognising them is what makes estimation and factorising quick. Squares to 15², cubes to 5³, and the square roots that follow from them.
Beyond that, the calculator handles it — but only if the expression is entered with the brackets the mathematics requires. √(9 + 16) is 5; √9 + 16 is 19. The root sign in handwriting extends over everything beneath it, and that grouping must be typed in as brackets.
Before you leave this chapter
- Multiplying adds indices, dividing subtracts, a power of a power multiplies — but only for the same base.
- a⁰ = 1 and a⁻ⁿ = 1/aⁿ, both forced by the division law.
- a^(m/n) means the n-th root, then the m-th power. Root first.
- Standard form is a × 10ⁿ with 1 ≤ a < 10. Both conditions are marked.
- The bar of a root sign is a bracket — type it as one.
Why exponential growth outruns everything
A quantity that grows by a fixed amount each step follows a straight line. One that grows by a fixed factor follows an exponential curve, and the difference between them is enormous over any length of time.
Doubling repeatedly is the clearest case. Start with one rupee and double it every day: after ten days it is Rs 1024, after twenty it is over a million, after thirty it is more than a billion. Nothing about the first few days suggests where it ends up, which is why exponential growth is so consistently underestimated.
Switch between Linear, Quadratic and Exponential. Near the origin the exponential looks unremarkable; a little further out it has left the others behind entirely, and no polynomial ever catches it.
The rule of 72
A quantity growing at r% per period roughly doubles every 72 ÷ r periods. At 8% that is about 9 years; at 12%, about 6. It is an approximation rather than a formula to quote in an answer, but it is a fast sanity check on any compound growth calculation — and it works because of the index laws in this chapter.