Laws of Indices
The laws of indices are rules for simplifying expressions where the same base is multiplied, divided or raised to a power.
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The laws of indices are rules for simplifying expressions where the same base is multiplied, divided or raised to a power.
- Multiply: add the powers
- Divide: subtract the powers
- Power of a power: multiply the powers
🧠 Quick quiz
Three questions. Score 2 or more to complete the topic.
What is Laws of Indices?
Within Mathematics, which covers numbers, algebra, geometry, trigonometry and statistics from first principles to exam level, this topic explains Laws of Indices. Maths builds step by step, so each topic here gives you a rule, a picture and a quick check you can use on homework and in exams.
Check that you can explain Laws of Indices in your own words and give one example from everyday life. If you can do that, you have understood the core idea.
Understanding Laws of Indices gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
Laws of Indices in detail
The laws of indices let you simplify expressions with the same base. To multiply, add the powers: aᵐ × aⁿ = aᵐ⁺ⁿ. To divide, subtract them: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. To raise a power to a power, multiply them: (aᵐ)ⁿ = aᵐⁿ.
These rules only work when the bases are the same, so 2³ × 3² cannot be combined this way. They are used to simplify algebra, to solve equations where the unknown is an exponent and to understand why a⁰ = 1. Practising with numbers first makes the algebraic versions easier to trust.
How the interactive helps you understand Laws of Indices
Click a card to flip it and reveal the answer, then move on to the next one.
Before you change anything, predict the result. Then change one value at a time and check whether your prediction was right. This is a quick way to build a correct mental model of Laws of Indices.
Real-life uses and examples
- Use Laws of Indices when comparing prices, discounts or recipe quantities in everyday life.
- Explain Laws of Indices to a friend with a simple picture, such as a row of tiles or a bag of sweets.
- Spot Laws of Indices in exam questions, where it often sits inside a short word problem.
Common mistakes to avoid
- Forgetting units, signs or brackets when you apply Laws of Indices.
- Memorising the steps for Laws of Indices without checking why they work.
- Learning the definition without testing it. Use the quiz and the interactive to check that you can apply the idea.
Frequently asked questions about Laws of Indices
What is Laws of Indices?
The laws of indices are rules for simplifying expressions where the same base is multiplied, divided or raised to a power.
What are the most important points about Laws of Indices?
The main points are: Multiply: add the powers; Divide: subtract the powers; Power of a power: multiply the powers.
Why is Laws of Indices useful?
Understanding Laws of Indices gives you a reliable method you will reuse in later maths units, in science lessons and in data work.
How can I practise Laws of Indices?
Open the interactive at the top of this page and change the controls one at a time. Predict the result before each change, then take the three-question quiz. A score of two or more marks the topic as complete on this device.
Which topics are related to Laws of Indices?
Related topics in Mathematics include Standard Form, Significant Figures, Sequences.