Number
Powers and Roots
Year 7 · Year 8 · Year 9
- ✓By the end of this lesson students will be able to understand and use index notation for powers.
- ✓By the end of this lesson students will be able to apply the index laws for multiplication, division, and raising a power to a power.
- ✓By the end of this lesson students will be able to calculate square roots and cube roots of positive integers.
- ✓By the end of this lesson students will be able to recognise and recall common square numbers and cube numbers.
- ✓By the end of this lesson students will be able to simplify expressions involving powers and roots.
Key concepts
Index notation is a way of writing repeated multiplication. For example, 2 × 2 × 2 × 2 can be written as 2⁴. In a power like aⁿ, 'a' is called the base and 'n' is called the index (or power or exponent). It means 'a' multiplied by itself 'n' times.
A square number is the result of multiplying an integer by itself. For example, 3² = 3 × 3 = 9. We say '3 squared' or '3 to the power of 2'.
A cube number is the result of multiplying an integer by itself three times. For example, 2³ = 2 × 2 × 2 = 8. We say '2 cubed' or '2 to the power of 3'.
The square root of a number is a value that, when multiplied by itself, gives the original number. The symbol for square root is √. For example, √25 = 5 because 5 × 5 = 25. Every positive number has two square roots, one positive and one negative, but at Key Stage 3 we usually focus on the positive root.
The cube root of a number is a value that, when multiplied by itself three times, gives the original number. The symbol for cube root is ∛. For example, ∛8 = 2 because 2 × 2 × 2 = 8.
When multiplying powers with the same base, you add the indices.
When dividing powers with the same base, you subtract the indices.
When raising a power to another power, you multiply the indices.
Key facts to remember
- 1aⁿ means 'a' multiplied by itself 'n' times. 'a' is the base, 'n' is the index (or power/exponent).
- 2Any number to the power of 1 is itself: a¹ = a.
- 3Any non-zero number to the power of 0 is 1: a⁰ = 1.
- 4Index Law 1: aᵐ × aⁿ = a⁽ᵐ⁺ⁿ⁾ (When multiplying powers with the same base, add the indices).
- 5Index Law 2: aᵐ ÷ aⁿ = a⁽ᵐ⁻ⁿ⁾ (When dividing powers with the same base, subtract the indices).
- 6Index Law 3: (aᵐ)ⁿ = a⁽ᵐ×ⁿ⁾ (When raising a power to another power, multiply the indices).
- 7Common square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
- 8Common cube numbers: 1, 8, 27, 64, 125.
Worked examples
Example 1
Simplify 3⁵ × 3².
Answer
3⁷
You do not need to calculate the final value of 3⁷ unless asked.
Example 2
Simplify (x⁴)³ ÷ x².
Answer
x¹⁰
Always deal with brackets first when simplifying expressions involving powers.
Example 3
Calculate √81 + ∛27.
Answer
12
It is helpful to memorise common square and cube numbers.
Common mistakes
- ✗Multiplying the bases when applying the multiplication law: e.g., 2³ × 2⁴ ≠ 4⁷. The base stays the same.
- ✗Confusing 2³ with 2 × 3. 2³ = 2 × 2 × 2 = 8, whereas 2 × 3 = 6.
- ✗Incorrectly stating that a⁰ = 0. Remember, any non-zero number to the power of 0 is 1.
- ✗Adding or subtracting indices when they should be multiplied, or vice versa, especially with (aᵐ)ⁿ.
- ✗Confusing square roots (√) with cube roots (∛).
Exam tips
- ★Always show your working when applying index laws. This helps you to gain method marks even if your final answer is incorrect.
- ★Memorise the first few square and cube numbers. This will speed up calculations and help you recognise them quickly.
- ★Pay close attention to the base. Index laws only apply when the bases are the same.
- ★Be careful with notation: ensure you use the correct symbol for square root (√) and cube root (∛).
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