IB Maths AI SL Topic 1 — Number & Algebra Paper 1 & 2 Must memorise ~7 min read

Laws of Indices

Indices (powers) come up everywhere in AI SL — standard form, exponential growth, compound interest, finance models. The good news: there are just 8 rules to memorise, and they all do one of three jobs: combine powers (× and ÷), simplify brackets, or flip negatives. None of them are in the formula booklet, so you have to know them cold.

📘 The 8 rules — memorise these

Multiply (same base): x^m × xⁿ = x^m+n — add the powers
Divide (same base): x^m ÷ xⁿ = x^m−n — subtract the powers
Power of a power: (x^m)ⁿ = x^mn — multiply the powers
Power of a product: (xy)^m = x^m y^m — share the power across
Power of a quotient: (x/y)^m = x^m/y^m — share the power top and bottom
Power of 1: x¹ = x — any number to the 1 is itself
Power of 0: x⁰ = 1 (for x ≠ 0) — any non-zero number to the 0 is just 1
Negative power: x^−m = 1/x^m — flip it into a fraction

AI SL note: you do NOT need fractional indices (like x^1/2 = √x). Those are HL-only. For AI SL, the 8 rules above are all you need.

The 8 rules — your cheat sheet

The 4 teal cards are the workhorse rules (combining powers). The 4 orange cards are the special-case rules — easy to forget under exam pressure. Drill these until they’re automatic.

Changing the base — when bases look different but aren’t

The index rules only work when the bases match. So 2³ × 5² cannot be combined — different bases. But sometimes bases LOOK different but secretly are the same. For example, 4 = 2² and 8 = 2³, so anything involving 2, 4, 8, 16, 32, … can be rewritten as powers of 2.

The trick Rewrite each base as a power of the smallest common base, then use the rules.

Quick reference for AI SL bases that often appear:

Powers of 2: 2, 4 = 2², 8 = 2³, 16 = 2⁴, 32 = 2⁵, 64 = 2⁶
Powers of 3: 3, 9 = 3², 27 = 3³, 81 = 3⁴
Powers of 5: 5, 25 = 5², 125 = 5³
Powers of 10: 10, 100 = 10², 1000 = 10³ (these come up in standard form too)

🧭 Recipe — simplifying any expression with indices

Deal with brackets first: if you see (something)ⁿ, share the power across using rules 3, 4, or 5.
Check the bases: do they all match? If not, try to rewrite them as powers of the same base (e.g., 8 = 2³).
Combine using × and ÷: for the same base, ADD powers when multiplying, SUBTRACT powers when dividing.
Handle negative powers: x^−m = 1/x^m. If the question asks for a positive index in the final answer, flip negatives into the denominator.
Tidy up coefficients separately: numbers in front of the variable (like the 4 in 4x³) multiply or divide as ordinary numbers — separate from the index work.

Worked examples

WE 1

Multiplying with the same base

Simplify 5³ × 5⁴, giving your answer as a single power of 5 and as a single number.

Step 1 — same base (5), so ADD the powers (Rule 1) 5³ × 5⁴ = 5³⁺⁴ = 5⁷ Step 2 — evaluate as a number 5⁷ = 5 × 5 × 5 × 5 × 5 × 5 × 5 = 78 125 5⁷ = 78 125 a common slip: multiplying the powers instead of adding them. 5³ × 5⁴ is NOT 5¹². You ADD when multiplying same bases.

WE 2

Dividing with the same base

Simplify 2⁸ ÷ 2³, giving your answer as a single number.

Step 1 — same base (2), so SUBTRACT the powers (Rule 2) 2⁸ ÷ 2³ = 2⁸⁻³ = 2⁵ Step 2 — evaluate 2⁵ = 2 × 2 × 2 × 2 × 2 = 32 2⁸ ÷ 2³ = 32 order of subtraction matters: top power minus bottom power. 2⁸ ÷ 2³ = 2⁵, NOT 2⁻⁵. If you reverse it you’ll get a negative answer.

WE 3

Negative power — express as a fraction

Evaluate 3⁻² as a fraction.

Step 1 — apply the negative-power rule (Rule 8) 3⁻² = 13² Step 2 — evaluate the bottom 3² = 9 3⁻² = 19 negative does NOT mean “make it negative”. 3⁻² = 1/9, which is positive. The minus sign tells you to FLIP it into a fraction — it doesn’t change the sign of the answer.

WE 4

Brackets — power of a product

Simplify (4x³)².

Step 1 — share the power across both parts inside (Rule 4) (4x³)² = 4² × (x³)² Step 2 — work each part 4² = 16 (x³)² = x³ˣ² = x⁶ (Rule 3: multiply powers) Step 3 — combine (4x³)² = 16x⁶ don’t forget to square the 4! Writing 4x⁶ is a very common mistake — the outside power applies to EVERYTHING inside the brackets, including the coefficient.

WE 5

Mixed: multiply, then divide, two variables

Simplify a⁵b² × a²b³a⁴b.

Step 1 — work the top first: multiply (add powers, per variable) a⁵ × a² = a⁵⁺² = a⁷ b² × b³ = b²⁺³ = b⁵ top = a⁷ b⁵ Step 2 — now divide by the bottom (subtract powers) a⁷ ÷ a⁴ = a⁷⁻⁴ = a³ b⁵ ÷ b¹ = b⁵⁻¹ = b⁴ (remember: b alone means b¹) a³ b⁴ handle a’s and b’s SEPARATELY — they’re different bases, so they don’t mix. Treat each variable as its own little problem, then write the answers side by side.

WE 6

Changing the base

Express 9² × 27³ as a single power of 3.

Step 1 — spot the common base 9 = 3², 27 = 3³ Step 2 — rewrite both numbers using base 3 9² = (3²)² = 3²ˣ² = 3⁴ (Rule 3) 27³ = (3³)³ = 3³ˣ³ = 3⁹ (Rule 3) Step 3 — multiply (same base now, ADD powers) 3⁴ × 3⁹ = 3⁴⁺⁹ = 3¹³ 9² × 27³ = 3¹³ without changing the base you’d be stuck — 9 and 27 are different bases. Spotting that both are powers of 3 is the whole trick. Practice this: 4 = 2², 8 = 2³, 16 = 2⁴ — these come up constantly.

💡 Top tips

Same base first — always check: index laws only work when bases match. 2³ × 3² cannot be combined; 2³ × 2² can. If bases don’t match, try to change the base.
“x alone” means x¹: if you see a variable with no visible power, the power is 1. Easy to forget when subtracting (e.g., b⁵ ÷ b = b⁵⁻¹ = b⁴, not b⁵).
Brackets share the power with EVERYTHING inside: (2x)³ = 8x³ (not 2x³). The 2 gets cubed too.
Negative exponents are just fractions: x⁻² isn’t “negative” — it’s 1/x². Always positive in the end (if x > 0).
Anything to the 0 is 1: 7⁰ = 1, 1000⁰ = 1, (3x²)⁰ = 1. This even works when you spot it in the middle of a problem — replace the whole thing with 1 and carry on.

⚠ Common mistakes

Multiplying powers when you should add: 5³ × 5⁴ = 5⁷ (add), NOT 5¹² (multiply). You ONLY multiply powers when you have a power of a power: (5³)⁴ = 5¹².
Forgetting to apply the outside power to the coefficient: (4x³)² = 16x⁶, NOT 4x⁶. The 4 gets squared as well — it’s inside the brackets.
Treating x⁻² as a negative number: x⁻² is a fraction (1/x²), not a negative quantity. 2⁻² = 1/4, which is positive.
Mixing different bases: 2³ + 3² is NOT 5⁵ or 6⁵ or anything similar — you just have to evaluate each separately (= 8 + 9 = 17). Index laws don’t apply across different bases.
Combining + and × in the same step: index laws are for × and ÷ ONLY. (x + y)² is NOT x² + y² (you’d need to expand the bracket).

Up next: Approximation. Now that you can handle big and small numbers cleanly, the next skill is rounding them sensibly — to a number of significant figures or decimal places — and knowing when to round UP regardless (think: how many buses for 47 students). Quick, practical, full of “exam-trap” examples.

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