√144 at a glance
- Exact value
- 12
- Decimal (10 places)
- 12
- Rounded
- 12
- Perfect square?
- Yes — 12²
- Rational?
- Rational (whole number)
- Both square roots
- ±12
- Prime factorization
- 2⁴ × 3²
- Cube root
- 5.241483
How to simplify √144
Write 144 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√144 = 2² × 3 = 12
144 has 8 factor pairs (1 × 144, 2 × 72, 3 × 48, 4 × 36, 6 × 24, 8 × 18, 9 × 16, 12 × 12). The pair with two equal numbers, 12 × 12, is what makes it a perfect square — and why it has an odd number of factors.
Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 144 = 1 + 3 + 5 + 7 + … + 23, the first 12 odd numbers.
Where √144 sits between perfect squares
144 is 12². The perfect squares on either side are 121 (11²) and 169 (13²), so the gaps are 23 below and 25 above — consecutive squares always differ by consecutive odd numbers.
Finding √144 with the Babylonian method
Picture a rectangle with an area of 144 and one side x; the other side must be 144 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √144.
To show the method at work on a perfect square, start one above the answer, at 13:
| Step | Guess x | 144 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 11.0769230769 | 12.0384615385 | 1 |
| 2 | 12.0384615385 | 11.9616613419 | 12.0000614402 | 4 |
| 3 | 12.0000614402 | 11.9999385602 | 12.0000000002 | 9 |
| 4 | 12.0000000002 | 11.9999999998 | 12.0000000000 | all 10 shown |
The count of correct decimals went 1, 4, 9 and all 10 over 4 steps — roughly doubling each time — until the guess matched √144 = 12.0000000000 to every decimal shown.
√144 in geometry and everyday measurements
- A square room or garden bed covering 144 square feet measures exactly 12 ft (12 ft) along each wall.
- 144 is not a sum of two whole-number squares — no pair of squares adds up to it — so √144 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 8 × 8 box, because 4² + 8² + 8² = 144.
Square roots near √144 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √141 | √141 | 11.8743 | No |
| √142 | √142 | 11.9164 | No |
| √143 | √143 | 11.9583 | No |
| √144 | 12 | 12.0000 | Yes |
| √145 | √145 | 12.0416 | No |
| √146 | √146 | 12.0830 | No |
| √147 | 7√3 | 12.1244 | No |
- The cube root of 144 is about 5.241483.
- Taking the square root twice gives the fourth root: √12 ≈ 3.464102.
- Four times the radicand doubles the root: √576 = 2 × √144 ≈ 24.
Frequently asked questions
What is the square root of 144?
The square root of 144 is 12, because 12 × 12 = 144. Strictly, 144 has two square roots, 12 and −12; the √ symbol means the positive one.
Is the square root of 144 rational or irrational?
Rational. 144 is a perfect square, so its square root is the whole number 12.
Can √144 be simplified?
Yes, all the way to a whole number: √144 = 12.
What is √144 rounded to two decimal places?
√144 is exactly 12, or 12.00 written to two decimal places.