Free IB Maths AA HL lesson: Laws of indices (all real exponents)
A ready-to-teach IB Maths AA HL lesson on Laws of indices (all real exponents) (Unit 1 · Number & Algebra): editable slides and teacher notes, free; a printable worksheet in 3 levels with answers for school plans. About 55 minutes.
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What's inside
- Slides (PowerPoint, editable) · freeabout 12 slides: do now, objectives, key ideas, a worked example step by step, “your turn”, practice QR code, exit ticket
- Teacher notes (PDF) · freelesson plan with timings, common mistakes, homework links
- Worksheet (PDF) · school plans6 questions, 22 marks: Fluency, Reasoning, Exam-style. Included with school plans and the teacher trial.
- Answers (PDF) · school plansthe answer to every worksheet question, from its mark scheme. Included with school plans and the teacher trial.
Learning objectives
- Laws of exponents (all real powers)
Key ideas on the slides
- aᵐ aⁿ = aᵐ⁺ⁿ, (aᵐ)/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿ bⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(m/n) = ⁿ√(aᵐ).
- Write every term with the same base (usually a prime) before combining: (8ˣ × 4)/(2ˣ⁺¹) = (2³ˣ × 2²)/(2ˣ⁺¹) = 2²ˣ⁺¹.
- An equation such as 9ˣ = 27ˣ⁻¹ becomes 3²ˣ = 3³ˣ⁻³, so 2x = 3x − 3 and x = 3.
- A hidden quadratic: 4ˣ − 6 × 2ˣ + 8 = 0 is (2ˣ)² − 6(2ˣ) + 8 = 0, so 2ˣ = 2 or 4.
By the end of the lesson, students should be able to answer a question like: Simplify the expression (16x^4)^(1/2) divided by x^-2.
Key vocabulary
- exponent: The power a base is raised to.
- base: The number being raised to a power.
- hidden quadratic: An equation that is quadratic in an expression such as 2ˣ.
- index: The small raised number showing how many times the base is multiplied by itself (plural: indices).
Common mistakes this lesson tackles
- Logs and exponentials: hidden quadratics and misapplied log laws. Look for aˣ and a²ˣ together: substitute u = aˣ. Logs of a sum do not split. log(xy)ⁿ = n log(xy) = n(log x + log y).
- Multiplying bases: 2³ × 3² ≠ 6⁵.
- Treating (a + b)² as a² + b².
- Dropping solutions such as 2ˣ = −1 without saying why (it has no real solution).
Calculator tips
- Graph both sides and use intersect to check your exact answers.
- Check an index law with numbers on the calculator, e.g. 2³ × 2⁴ against 2⁷.
Lesson plan
| 0–5 min | Do now: students write down what they remember about Infinite geometric series (|r| < 1) and Sigma notation |
| 5–8 min | Share the learning objective and success criteria |
| 8–20 min | Teach: the worked example, step by step; students copy the method |
| 20–25 min | Your turn: 1 mini-whiteboard question; check every board before moving on |
| 25–42 min | Practise: the worksheet (6 questions in 3 levels, with school plans and the teacher trial) or the topic's practice questions online; move up a level after two right in a row |
| 42–47 min | Plenary: 1-question exit ticket; collect it to plan the next lesson |
For students: practise this topic
The worksheet's QR code opens the IB Maths AA HL Unit 1: Exponentials and Logarithms practice questions.
What schools get
- Set this lesson as homework, marked by AI against the mark scheme, with every result in your markbook (with a School plan).
- This lesson's worksheet in three levels, with its answers, ready to print.
- Support, Core and Stretch versions and A/B copies of the worksheet from the full question bank, in the worksheet builder.
- Create lesson: this lesson rebuilt for your class's next scheme-of-work lesson, from the full bank.
School licences from €490 a year. Try everything free for 30 days as a teacher: no card.
Questions teachers ask
Is this lesson free?
The slides and the teacher notes are free to download with a free IB Math Revision teacher account (sign in with Google, Microsoft or email); no card. The printable worksheet in three levels and its answers are included with school plans and the free 30-day teacher trial.
Can I edit the slides?
Yes. The slides are an ordinary PowerPoint file: change, add or delete anything. They also open in Google Slides and Keynote. The logo and web address sit on the slide master, so your edits do not move them.
What do students use for homework?
The practice slide's QR code and web address take students to the IB Maths AA HL Unit 1: Exponentials and Logarithms practice questions on ibmathrevision.com, where the free questions are open and the full bank comes with a plan.