IB Maths AA HLTopic 1 — Number & AlgebraPaper 1 & 2~8 min read
Laws of Indices
“Indices” is just IB-speak for “powers” or “exponents”. The laws below sit underneath nearly every algebraic simplification you’ll meet — from solving exponential equations, to differentiating, to handling logs. None of them are in the formula booklet, so they have to live in your head. Don’t worry though — the rules are short, and once you spot the pattern, the rest follows automatically.
📘 What you need to know
The laws only work when the bases match. 23 × 52 can’t be simplified by indices alone.
Same base, multiplying: add the powers. Same base, dividing: subtract the powers.
Power of a power: multiply the powers. (32)4 = 38.
Negative powers mean reciprocals: a−n = 1 / an.
Fractional powers mean roots: a1/n = n√a.
If bases don’t match, see if you can rewrite one as a power of the other (changing the base).
Not in the formula booklet — memorise.
The five core rules
If you only memorise five things from this whole topic, make it these. They show up in every paper.
Rule
In plain English
Quick example
am × an = am+n
Same base, multiplying — add the powers.
52 × 54 = 56
am ÷ an = am−n
Same base, dividing — subtract the powers.
97 ÷ 93 = 94
(am)n = amn
Power of a power — multiply the powers.
(32)4 = 38
a0 = 1
Anything (non-zero) raised to the power 0 equals 1.
170 = 1
a1 = a
Anything raised to the power 1 is itself.
61 = 6
🤔 Why is anything to the power 0 equal to 1?
It falls out of the divide-rule. Take 53 ÷ 53. By cancellation, that’s 1. By the rule, it’s 53−3 = 50. So 50 must equal 1. Same logic works for any non-zero base.
Distributing across products and fractions
When the thing inside the bracket is a product or a quotient, the power applies to each piece.
Rule
In plain English
Quick example
(ab)n = anbn
A product to a power — apply the power to each factor.
(2 × 7)2 = 22 × 72
(ab)n = anbn
A fraction to a power — apply the power to top and bottom.
(52)2 = 254
These rules do not work for sums. (a + b)2 is not a2 + b2. That’s the most expensive mistake at SL/HL — easily caught by trying numbers: (3 + 4)2 = 49, but 32 + 42 = 25.
Negative powers — flip into a fraction
A negative power means “take the reciprocal”. That’s it.
Negative powera−n = 1an✗ not in the formula booklet
Quick examples: 4−1 = 14 | 2−3 = 18 | x−5 = 1x5
For a fraction with a negative power, the same idea gives a clean shortcut: flip the fraction and make the power positive.
Fraction with negative power — flip
(ab)−n = (ba)n
Step 1: Take the 5th root first (the friendlier route)321/5 = 5√32 = 2Step 2: Now apply the power 2322/5 = (321/5)2 = 22= 4root-first beats power-first here — 32² = 1024 is a calculator detour
WE 3
Negative and fractional power together
Evaluate 64−1/3, giving your answer as a fraction.
Step 1: Negative power → reciprocal64−1/3 = 1641/3Step 2: Fractional power → cube root641/3 = 3√64 = 4= 14handle the sign of the power first, then the size
WE 4
Algebraic simplification
Simplify (2x4y)38x5y2, giving your answer with positive indices only.
Step 1: Expand the numerator using (ab)n = anbn(2x4y)3 = 23x12y3 = 8x12y3Step 2: Divide by the denominator= 8x12y38x5y2= x12−5y3−2= x7y
WE 5
Algebraic — finish with positive indices
Simplify (3x−2y4)2 × (x3y)−1, giving your answer with positive indices only.
Step 1: Distribute the outer powers(3x−2y4)2 = 9x−4y8(x3y)−1 = x−3y−1Step 2: Multiply — same base, add powers= 9 × x−4+(−3) × y8+(−1)= 9x−7y7Step 3: Rewrite negative powers as fractions= 9y7x7
WE 6
Solve an equation by changing the base
Solve 4x = 8 for x.
Step 1: Both sides are powers of 24 = 22, 8 = 23Step 2: Rewrite the equation(22)x = 2322x = 23Step 3: Same base — equate the powers2x = 3x = 32no logs needed — change of base does the whole job
💡 Top tips
Always check the bases match. Index laws are useless on 23 × 52 — first ask whether you can rewrite one base.
Root-first for fractional powers — when the number under the root is a perfect nth power, take the root first and you avoid huge intermediate numbers.
If a question says “give your answer with positive indices“, that’s a deliberate instruction. Negative powers must be flipped into fractions in the final line.
Treat the negative sign carefully. 2−3 means 18, not −8. The negative is in the exponent, not the base.
For algebra, deal with the numerical coefficients first, then the variables. Splitting them keeps the working clean.
For equations, change of base > logs whenever the numbers are friendly. Logs are the backup, not the first tool.
The laws are not in the formula booklet — practice with mixed problems until they’re automatic.
⚠ Common mistakes
Adding powers when you should multiply (or vice versa). Multiplying same-base terms → add. Power of a power → multiply. The two are easy to swap under pressure.
Distributing the power across a sum. (a + b)2 ≠ a2 + b2. The product/quotient laws don’t apply to addition.
Treating 2−3 as a negative number. It’s a positive fraction: 18. Negative power ≠ negative value.
Forgetting to apply the outer power to the coefficient. (3x2)3 = 27x6, not 3x6. The 3 gets cubed too.
Mixing up a0 and 0.a0 = 1 (when a ≠ 0). The “1” is what surprises people.
Skipping the final positive-indices step. Leaving x−2 when the question asked for positive indices loses you a mark, even if the answer is “right”.
Bases that aren’t actually equal. 43 and 23 are not the same base — rewrite 4 = 22 first.
Index laws are the bedrock of nearly everything in algebra and calculus — exponential/log questions, polynomial differentiation, integration of xn, even basic equation solving. The earlier you get fluent, the easier later topics become.
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