Square Roots
“A square root is a number with the square 'undone'.”
The formula
√a = b ⇔ b² = a (a ≥ 0, b ≥ 0)How to read it: the non-negative number b that squares to a is the square root √a
- √a
- — the non-negative number that squares to a (the positive square root)
- b² = a
- — squaring b brings back the original number a
- √
- — the radical — the symbol for 'undo the square'
The hook
A square root runs squaring in reverse — it asks 'what do I square to get this number?'
In plain words
When some number squared gives a, that number is a square root of a. √a is the one that is 0 or positive.
The intuition
Picture a square with area a. The length of one side is exactly √a. A square of area 9 has a side of 3, so √9 = 3.
How it's built
Squaring (□²) is the forward move; taking a root (√□) undoes it. Being inverses, they cancel: (√a)² = a and √(b²) = b (for b≥0).
Example
Take √9. Since 3² = 9, √9 = 3 — the same as the side of a square with area 9 being 3.
Common mistake
√9 is 3, not ±3. The radical √ points only to the non-negative value. Two numbers square to 9 (±3), but the symbol √9 means the single value 3.
Where it's used
Finding distances with the Pythagorean theorem, side lengths of shapes, standard deviation, and many physics formulas — anywhere you must undo a square.
Where it came from
The radical √ is said to come from a stylized r for the Latin radix ('root'), carrying the sense of digging out a number's 'root'.
Prerequisites
Quick check
What is √16?
- 2
- 4✓
- 8
- 256
Practice
What is √36?
Answer: 6
- Find the non-negative number that squares to 36
- 6² = 36 → √36 = 6
Takeaway: Find the value that squares to the number.
What is √81?
Answer: 9
- 9² = 81
- → √81 = 9
Takeaway: Knowing your squares reveals the square roots.
What is (√7)²?
Answer: 7
- Squaring and square-rooting cancel each other
- (√7)² = 7
Takeaway: Squaring a square root gives back the number inside.
Between which two integers does √10 lie? Write why.
Answer: undefined
- 3² = 9, 4² = 16
- 9 < 10 < 16, so 3 < √10 < 4
- √10 is between 3 and 4 (about 3.16)
Takeaway: Estimate a root using the nearest perfect squares.