√49 at a glance
- Exact value
- 7
- Decimal (10 places)
- 7
- Rounded
- 7
- Perfect square?
- Yes — 7²
- Rational?
- Rational (whole number)
- Both square roots
- ±7
- Prime factorization
- 7²
- Cube root
- 3.659306
How to simplify √49
Write 49 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√49 = 7 = 7
49 has 2 factor pairs (1 × 49, 7 × 7). The pair with two equal numbers, 7 × 7, 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 49 = 1 + 3 + 5 + 7 + … + 13, the first 7 odd numbers.
Where √49 sits between perfect squares
49 is 7². The perfect squares on either side are 36 (6²) and 64 (8²), so the gaps are 13 below and 15 above — consecutive squares always differ by consecutive odd numbers.
Finding √49 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 49: following the tangent line down to zero simplifies to averaging x with 49 ÷ x.
To show the method at work on a perfect square, start one above the answer, at 8:
| Step | Guess x | 49 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 8.0000000000 | 6.1250000000 | 7.0625000000 | 1 |
| 2 | 7.0625000000 | 6.9380530973 | 7.0002765487 | 3 |
| 3 | 7.0002765487 | 6.9997234623 | 7.0000000055 | 8 |
| 4 | 7.0000000055 | 6.9999999945 | 7.0000000000 | all 10 shown |
The count of correct decimals went 1, 3, 8 and all 10 over 4 steps — roughly doubling each time — until the guess matched √49 = 7.0000000000 to every decimal shown.
√49 in geometry and everyday measurements
- A square room or garden bed covering 49 square feet measures exactly 7 ft (7 ft) along each wall.
- 49 is not a sum of two whole-number squares — no pair of squares adds up to it — so √49 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 6 box, because 2² + 3² + 6² = 49.
Square roots near √49 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √46 | √46 | 6.7823 | No |
| √47 | √47 | 6.8557 | No |
| √48 | 4√3 | 6.9282 | No |
| √49 | 7 | 7.0000 | Yes |
| √50 | 5√2 | 7.0711 | No |
| √51 | √51 | 7.1414 | No |
| √52 | 2√13 | 7.2111 | No |
- The cube root of 49 is about 3.659306.
- Taking the square root twice gives the fourth root: √7 ≈ 2.645751.
- Four times the radicand doubles the root: √196 = 2 × √49 ≈ 14.
Frequently asked questions
What is the square root of 49?
The square root of 49 is 7, because 7 × 7 = 49. Strictly, 49 has two square roots, 7 and −7; the √ symbol means the positive one.
Is the square root of 49 rational or irrational?
Rational. 49 is a perfect square, so its square root is the whole number 7.
Can √49 be simplified?
Yes, all the way to a whole number: √49 = 7.
What is √49 rounded to two decimal places?
√49 is exactly 7, or 7.00 written to two decimal places.