Mistakes · Basic Math
5 Common Square Root Mistakes
Four of the five are caught by comparing your answer's decimal with the original. It takes five seconds.
1. Distributing over addition
The worst of the common square root mistakes, because the false version looks so reasonable. Distributing over addition is not allowed: √(9 + 16) is √25, which is 5 — not 3 + 4.
Roots distribute over multiplication and division and never over addition or subtraction. √(9 × 16) really is √9 × √16 = 12; √(9 + 16) has no such split.
2. Not simplifying fully
√72 = 2√18 is true and unfinished, because 18 still contains the square 9. The answer is 6√2, and a partly simplified surd usually scores as an incomplete answer.
- √72 = 2√18correct but not finished
- 2√18 = 2 · 3√218 = 9 · 2
- 6√2now the radicand has no square factor
The finish line is precise: the radicand must have no perfect square factor above 1. Test it against 4, 9, 25 and 49 before writing the final line.
3. Losing the negative root when solving
√49 is 7, because the symbol means the principal root. But x² = 49 has two solutions, 7 and −7, and writing only the first discards half the answer.
| Written as | Means | Answer |
|---|---|---|
| √49 | the principal root | 7 |
| x² = 49 | an equation to solve | x = 7 or x = −7 |
| ±√49 | both roots requested | ±7 |
The distinction is between evaluating a symbol and solving an equation. Squaring throws away the sign, so undoing it has to consider both possibilities.
4. Rounding before the end
√2 is exact; 1.41 is not. Rounding early compounds through every later step, and in a long calculation the final answer can be wrong in the second decimal place.
Keep the surd until the last line, then round once to whatever precision was asked for. If the exact answer is 6√2 and the question wants two decimals, write 6√2 and then 8.49 — not 8.49 all the way through.
5. Cancelling a root out of part of a fraction, and rationalising badly
In (3 + √2)/2 the 2s do not cancel, because the √2 is only part of the numerator. Cancelling needs a factor of the whole top and the whole bottom.
The same error appears after rationalising: 3√2/2 cannot be reduced to 3√1 or to 3/2 · √2 written as 3√2/2 — that last one is already correct, and the temptation is to keep going and lose something.
Two smaller ones worth naming
- Adding radicals that do not match. √3 + √5 is not √8. Only identical surds combine, exactly as only like terms combine in algebra.
- Failing to rationalise when asked. 3/√2 is a correct value and an unfinished answer if the question wanted a rational denominator. Multiply top and bottom by √2 to reach 3√2/2.
Neither changes the value; both change whether the answer is accepted. It is worth a quick estimate in each case to confirm the value survived the rearranging.
The check that catches four of the five
Check the decimal. Work out the decimal value of your answer and the decimal value of what you started with, and estimate roughly what it should be first. They must match.
| Started with | Answer | Decimal check |
|---|---|---|
| √72 | 6√2 | 8.4853 = 8.4853 ✓ |
| √72 | 2√18 | 8.4853 = 8.4853 ✓ but not simplified |
| √(9+16) | 3 + 4 | 5 ≠ 7 ✗ |
The middle row is the useful warning: a decimal check confirms the value but not the form. For full simplification you still have to ask whether the radicand has a square factor left.
Questions about common square root mistakes
Why is √(9 + 16) not 3 + 4?
Because roots do not distribute over addition. √(9 + 16) is √25, which is 5. They do distribute over multiplication: √(9 × 16) is √9 × √16 = 12.
How do I know a root is fully simplified?
The number left under the radical must have no perfect square factor above 1. Test it against 4, 9, 25 and 49.
When do I need the ± sign?
When solving an equation such as x² = 49, where both 7 and −7 work. The √ symbol on its own means the positive root only.
Should I use the decimal or the exact form?
Keep the exact form until the final line. Rounding early compounds the error through every step that follows.