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
Answer exam-style 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.