What Is a Square Root?
A square root answers the question: "What number, multiplied by itself, gives me this?"
√25 = 5 because 5 × 5 = 25. √9 = 3 because 3 × 3 = 9. √100 = 10 because 10 × 10 = 100.
Why "square" root?
A square with area 25 has side length 5 (because 5 × 5 = 25). Finding the side length of a square from its area is finding a square root. The name comes from geometry.
Perfect squares
Numbers with whole-number square roots are called perfect squares:
| Number | Square Root |
|---|---|
| 1 | √1 = 1 |
| 4 | √4 = 2 |
| 9 | √9 = 3 |
| 16 | √16 = 4 |
| 25 | √25 = 5 |
| 36 | √36 = 6 |
| 49 | √49 = 7 |
| 64 | √64 = 8 |
| 81 | √81 = 9 |
| 100 | √100 = 10 |
| 121 | √121 = 11 |
| 144 | √144 = 12 |
Memorizing these is extremely helpful for Pythagorean theorem problems.
Non-perfect squares
√2 ≈ 1.414... The square root of a non-perfect-square is an irrational number, a decimal that never terminates or repeats.
To estimate: √50 is between √49 (= 7) and √64 (= 8), closer to 7. So √50 ≈ 7.1.
The relationship to exponents
Square roots undo squaring:
- 5² = 25, and √25 = 5
- Squaring and square-rooting are inverse operations
In exponent notation: √x = x^(1/2).
Common confusion
Thinking √25 = 12.5: They divide 25 by 2 instead of finding what number times itself equals 25. Square root is not "half of", it is "what times itself equals."
Forgetting negative square roots: 5 × 5 = 25 and (-5) × (-5) = 25. Both are square roots of 25. The symbol √ refers to the positive root only.
Related concepts
- Exponents: square roots undo squaring
- Pythagorean theorem: uses square roots to find side lengths
- Irrational numbers: non-perfect-square roots are irrational