Inverse operations and the relationships between operations
An inverse undoes the original operation. They come in pairs:
- Addition (+) and subtraction (−)
- Multiplication (×) and division (÷)
- Squaring (²) and square-rooting (√)
- Cubing (³) and cube-rooting (³√)
- Raising to a power (xⁿ) and the corresponding root (ⁿ√)
Every algebraic rearrangement and almost every calculator-check trick is built from this idea.
Why inverses matter
You'll use them constantly:
- Solving equations — "do the same thing to both sides" usually means apply the inverse.
- Rearranging formulae — make a different letter the subject.
- Checking answers — if 84 ÷ 7 = 12, then 12 × 7 should equal 84.
- Working backwards from an answer — function machines, finding original prices, etc.
Function machines — the visual model
A function machine is a chain of operations:
INPUT → ×3 → +5 → OUTPUT
To undo it, run the chain backwards, swapping each operation for its inverse and reversing the order:
OUTPUT → −5 → ÷3 → INPUT
Worked example: a number is multiplied by 3 then 5 is added; the answer is 23. Find the original number. Reverse the chain: 23 − 5 = 18; 18 ÷ 3 = 6.
Solving simple equations using inverses
To solve 4x + 7 = 23 algebraically:
- Subtract 7 (inverse of +7): 4x = 16
- Divide by 4 (inverse of ×4): x = 4
The order matters — always undo the operation that's furthest from the variable first.
Squaring and square-rooting — the two-solution trap
The square root of a positive number has two answers, one positive and one negative. If x² = 25, then x = ±5.
Calculator √ buttons return only the positive root, but if you're solving an equation, write both unless the context (e.g. a length) requires the positive one.
Worked example: x² + 4 = 53. Inverses give x² = 49, then x = ±7.
Relationships between operations
- Multiplication is repeated addition: 6 × 4 = 6 + 6 + 6 + 6.
- Division is the inverse of multiplication: 24 ÷ 6 = 4 because 6 × 4 = 24.
- Powers are repeated multiplication: 2⁵ = 2 × 2 × 2 × 2 × 2.
- Roots invert powers: ³√64 = 4 because 4³ = 64.
You can use these links to check facts you can't recall. Forgot 8 × 9? Use 8 × 10 − 8 = 80 − 8 = 72.
⚠Common mistakes— Common mistakes (examiner traps)
- Reversing inverses but forgetting to reverse the order. Trying to undo ×3 then +5 by doing ÷3 first instead of −5 first.
- Forgetting the negative root when square-rooting in equations.
- Treating subtraction as commutative. 8 − 3 ≠ 3 − 8. The inverse of "subtract 3" is "add 3", not "subtract from 3".
- Cancelling powers and roots that aren't inverses — e.g. cancelling a square root against a cube. Only matched pairs cancel.
- Applying inverses to only one side of an equation. Whatever you do, do it to both sides.
➜Try this— Quick check
A number is squared, then 7 is subtracted, giving 18. Find the possible original numbers. Reverse: 18 + 7 = 25; √25 = ±5. So the number is 5 or −5.
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