The big idea: When the base stays the same, multiplication adds powers and division subtracts powers.
- same base
- powers to combine
Worked example
Simplify (a) x3 × x5 (b) y7 ÷ y2
Step by step
- Add powers when multiplying.
- Subtract powers when dividing.
Final answer
x8 and y5
Common mistake: Do not multiply the bases here.
The base stays the same; only the powers change.
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The big idea: A power raised to another power means multiply the powers.
A bracket raised to a power applies to every factor inside.
- multiply the powers
Worked example
Simplify (a) (x4)3 (b) (2a)3
Step by step
- Multiply powers.
- Raise each factor inside the bracket.
Final answer
x12 and 8a3
Slow down on brackets: If a whole bracket is raised to a power, make sure the power applies to every factor inside the bracket.
See how examiners mark answers
Access past paper questions with model answers. Learn exactly what earns marks and what doesn't.
| Expression | Meaning |
|---|---|
| a⁰ | 1, provided a ≠ 0 |
| a⁻ⁿ | 1/aⁿ (reciprocal) |
Worked example — zero and negative exponents
Simplify:
(a) 5⁰
(b) x⁻³
Step by step
- Any non-zero number to the power 0 equals 1.
- A negative power means take the reciprocal.
Final answer
5⁰ = 1 and x⁻³ = 1/x³
Why negative powers matter: A negative exponent does not make the answer negative.
It tells you to write the term as a reciprocal (flip it to 1/something).
Classic trap: x⁻² is not −x². It means 1/x² — same value, just flipped.
🎯 IB-style worked example
Skills you'll use here:
- Evaluate (1.3)⁻ᵗ = 1 / (1.3)t — negative exponent means reciprocal.
- Solve for t by isolating the exponential term first, then taking logs.
- Spot the asymptote — as t grows, (1.3)⁻ᵗ → 0, so P approaches a constant.
Scenario: A wildlife reserve releases cranes back into the wild. After release, the number of cranes still on the reserve, P, is modelled by:
P = 1200 + 600(1.3)⁻ᵗ
where t is the number of days since release began (t ≥ 0).
Part (a) — find P at t = 0 and t = 4
Using P = 1200 + 600(1.3)⁻ᵗ:
(a)(i) Find P at t = 0 (the moment release starts).
(a)(ii) Find P after 4 days. [3 marks]
Step by step
- Part (i): substitute t = 0 into the model. Anything to the power 0 is 1:
- Part (ii): substitute t = 4 into the model:
- Evaluate (1.3)⁻⁴ on the GDC. Round at the end:
Final answer
(i) P = 1800 (ii) P ≈ 1410
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Part (b) — find when P drops below 1300
Using the same model:
(b) Find the time t at which the population first falls below 1300. [2 marks]
Step by step
- Set up the equation by replacing P with the threshold value 1300:
- Isolate the exponential — subtract 1200, then divide by 600:
- Take logs of both sides to bring the exponent down:
- Solve for t and evaluate on the GDC:
Final answer
t ≈ 6.83 days
GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
Part (c) — find the smallest possible P
(c) According to the model, what is the smallest population the reserve can have, as t grows very large? [1 mark]
Step by step
- Think about what happens to (1.3)⁻ᵗ as t grows. A negative exponent with a base > 1 means . As t increases, the denominator gets huge, so the whole fraction shrinks towards 0:
- Substitute that limit into the model:
- Interpret. The population never actually reaches 1200 — it just gets closer and closer (this is called an asymptote). So the smallest possible population predicted by the model is:
Final answer
P → 1200 (the horizontal asymptote — the model never goes below this)
Exam tips for exponential-decay models:
- Part (a) is marked per sub-part — write (i) and (ii) clearly as separate calculations.
- For part (b), always show the isolation step ((1.3)⁻ᵗ = something) before taking logs — examiners need to see your method.
- For part (c), don't just write the number — mention the asymptote or limit as t → ∞ to show you understand the model behaviour.
- Use (−) not subtract for negative exponents on the GDC, or you'll get a SYNTAX ERROR.
The big idea: In longer expressions, simplify step by step using the exponent laws in a sensible order instead of trying to do everything at once.
Worked example
Simplify:
Step by step
- Power of a power first.
- Then multiply same base terms.
- Then divide.
Final answer
x5
Best habit: Write one clean line for each exponent law you use.
This reduces sign mistakes and is easier to follow under exam pressure.