Indices & Standard Form — Year 9 Revision Notes
These notes cover the laws of indices with zero and negative powers, and standard form for both large and small numbers — all at Year 9 (Stage 9) level.
Laws of indices, including zero and negative
The laws of indices still apply with zero and negative powers: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and a⁰ = 1. A negative index means a reciprocal, so a⁻ⁿ = 1/aⁿ. For example, 5⁻² = 1/25.
Key Facts & Formulas
- aᵐ × aⁿ = aᵐ⁺ⁿ
- a⁰ = 1
- a⁻ⁿ = 1/aⁿ
Tips
- A negative index flips the term into a fraction; it does not make it negative.
- Apply the laws to the bases first, then simplify.
Standard form for large and small numbers
Standard form is A × 10ⁿ with 1 ≤ A < 10. A positive power of 10 gives a large number, and a negative power gives a small number. So 7 × 10⁵ = 700 000, while 7 × 10⁻⁵ = 0.00007.
Key Facts & Formulas
- A × 10ⁿ, 1 ≤ A < 10
- 3 × 10⁻⁴ = 0.0003
Tips
- Large number → positive power; small number → negative power.
- Always keep A between 1 and 10.
Comparing numbers in standard form
To compare numbers in standard form, look at the power of 10 first: the larger power means the larger number (for positive powers), and the more negative power means the smaller number. Only compare the values of A if the powers are equal.
Key Facts & Formulas
- Compare powers of 10 first
- Then compare A
Tips
- For small numbers, the most negative power is the smallest.
- Convert to ordinary numbers if you are unsure.
Revision Checklist
- I can use the laws of indices with zero and negative powers
- I understand that a negative index means a reciprocal
- I can write large and small numbers in standard form
- I can compare numbers written in standard form
Frequently Asked Questions
Is 7 × 10⁻³ bigger or smaller than 7 × 10⁻²?
It is smaller. A more negative power of 10 means a smaller number: 7 × 10⁻³ = 0.007, while 7 × 10⁻² = 0.07.
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