Inverse operations
Every arithmetic operation has an inverse — the operation that "undoes" it. Edexcel uses this concept implicitly throughout (rearranging formulae, solving equations) and explicitly in checking calculations.
The four inverse pairs
| Operation | Inverse |
|---|---|
| Addition (+) | Subtraction (−) |
| Multiplication (×) | Division (÷) |
| Squaring (x²) | Square root (√) |
| Cubing (x³) | Cube root (∛) |
Power and root inverses generalise: x^n and ⁿ√x. Functional inverses (like exp ↔ log) appear at A-Level.
Using inverses to check answers
If 285 ÷ 19 = 15, check by 15 × 19 = 285. ✓ If 4.7 + 2.8 = 7.5, check by 7.5 − 2.8 = 4.7. ✓
This is the recommended Paper 1 strategy when time allows: every long calculation can be sense-checked.
Solving simple equations using inverse operations
x + 7 = 23 ⇒ x = 23 − 7 = 16. (Subtraction is the inverse of addition.) 5x = 30 ⇒ x = 30 ÷ 5 = 6. (Division is the inverse of multiplication.) x² = 49 ⇒ x = ±7. (Square root has two answers.)
Multi-step inverse
To solve 3x + 7 = 22:
- Subtract 7 (inverse of +7): 3x = 15.
- Divide by 3 (inverse of ×3): x = 5.
This builds toward the formal "solve linear equations" topic A17.
Inverses with negatives
The inverse of "subtract 5" is "add 5". The inverse of "multiply by −2" is "divide by −2".
Edexcel exam tip
If asked "calculate, and check your answer using inverse operations", you must show the check. A correct answer with no check loses the C1 communication mark.
⚠Common mistakes— Common errors
- Treating "inverse of squaring" as "halving" rather than square root.
- Forgetting the ± when taking a square root: √49 in equation form is ±7.
- Order errors in multi-step inverse: must reverse BIDMAS — undo addition/subtraction last when isolating.
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