Number

Integers, Powers and Roots — Revision Notes

Key Definitions and Terminology

• **Integer**: A whole number that can be positive, negative, or zero (e.g. …, −3, −2, −1, 0, 1, 2, 3, …). Integers do **not** include fractions or decimals.
• **Power (Index/Exponent)**: A shorthand notation for repeated multiplication. In the expression aⁿ, the base is *a* and the power (or index) is *n*. For example, 2⁵ means 2 × 2 × 2 × 2 × 2.
• **Square Number**: The result of multiplying an integer by itself. The first few square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
• **Cube Number**: The result of multiplying an integer by itself three times. The first few cube numbers are 1, 8, 27, 64, 125, 216, …
• **Square Root (√)**: The inverse of squaring. √49 = 7 because 7² = 49. Note that every positive number has a positive and a negative square root (e.g. √25 = ±5), but by convention the √ symbol refers to the **positive** root unless stated otherwise.
• **Cube Root (∛)**: The inverse of cubing. ∛64 = 4 because 4³ = 64. Cube roots can be negative (e.g. ∛(−8) = −2).

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Main Concepts

1. Recognising and Using Integer Properties

• You must be able to identify whether a number is a positive integer, negative integer, or zero.
• Calculations with negative integers follow sign rules:
• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative
• The same rules apply to division.

2. Laws of Indices (Powers)

These laws are essential and apply when the base is the same:

| Rule | Law | Example |

|------|-----|---------|

| Multiplication | aᵐ × aⁿ = aᵐ⁺ⁿ | 3⁴ × 3² = 3⁶ |

| Division | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁷ ÷ 5³ = 5⁴ |

| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (2³)⁴ = 2¹² |

| Zero index | a⁰ = 1 | 7⁰ = 1 |

| Negative index | a⁻ⁿ = 1/aⁿ | 4⁻² = 1/16 |

| Fractional index | a^(1/n) = ⁿ√a | 8^(1/3) = ∛8 = 2 |

| Combined fractional index | a^(m/n) = (ⁿ√a)ᵐ | 27^(2/3) = (∛27)² = 9 |

3. Squares, Cubes and Their Roots

• You should **memorise** squares from 1² to 15² and cubes from 1³ to 5³ (and ideally 10³).
• Square roots and cube roots reverse these operations.
• On a non-calculator paper, you are expected to evaluate these from memory or by working backwards.

4. Order of Operations with Powers

• Powers are evaluated **before** multiplication, division, addition, and subtraction (BIDMAS/BODMAS).
• Example: 3 + 2⁴ = 3 + 16 = 19 (not 5⁴).

5. Using Powers in Problem Solving

• Powers appear in area/volume problems (e.g. area of a square = side², volume of a cube = side³).
• Understanding inverse operations allows you to find unknown side lengths from given areas or volumes.

6. Negative and Fractional Indices Combined

• A negative fractional index combines two rules. Always deal with the **root first**, then the power, then the reciprocal.
• a^(−m/n) = 1 / (ⁿ√a)ᵐ

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Worked Examples

Worked Example 1: Evaluating a Fractional Index

Evaluate 32^(3/5).

Step 1: Identify the root and the power. The denominator is 5, so take the 5th root. The numerator is 3, so cube the result.

Step 2: ⁵√32 = 2 (since 2⁵ = 32)

Step 3: 2³ = 8

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Worked Example 2: Negative Index

Evaluate (4/9)^(−1/2).

Step 1: The negative sign means take the reciprocal → (9/4)^(1/2)

Step 2: The power 1/2 means square root → √(9/4) = √9 / √4 = 3/2

Answer: 3/2 (or 1.5)

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Worked Example 3: Real-World Application

A cube has a volume of 343 cm³. Find the side length.

Side length = ∛343

Since 7 × 7 × 7 = 343, the side length = 7 cm.

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Exam Technique Tips

1. **Show every intermediate step when applying index laws.** Edexcel mark schemes typically award a **method mark (M1)** for correctly setting up the index law and a separate **accuracy mark (A1)** for the final answer. Writing 27^(2/3) = (∛27)² = 3² = 9 earns full marks; jumping straight to 9 risks losing the method mark if incorrect.
2. **Watch for "write down" vs. "evaluate" vs. "simplify" in the question wording.** If the question says *evaluate*, you must give a numerical answer (not leave it in index form). If it says *simplify*, leave your answer in index form (e.g. 3⁵). Misreading this instruction is a common cause of lost marks on Edexcel papers.

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*Remember: practise index law questions from past papers — they appear almost every series on both Paper 1 (non-calculator) and Paper 2 (calculator).*