√961 at a glance
- Exact value
- 31
- Decimal (10 places)
- 31
- Rounded
- 31
- Perfect square?
- Yes — 31²
- Rational?
- Rational (whole number)
- Both square roots
- ±31
- Prime factorization
- 31²
- Cube root
- 9.868272
How to simplify √961
Write 961 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√961 = 31 = 31
961 has 2 factor pairs (1 × 961, 31 × 31). The pair with two equal numbers, 31 × 31, 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 961 = 1 + 3 + 5 + 7 + … + 61, the first 31 odd numbers.
Where √961 sits between perfect squares
961 is 31². The perfect squares on either side are 900 (30²) and 1,024 (32²), so the gaps are 61 below and 63 above — consecutive squares always differ by consecutive odd numbers.
Finding √961 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 961: following the tangent line down to zero simplifies to averaging x with 961 ÷ x.
To show the method at work on a perfect square, start one above the answer, at 32:
| Step | Guess x | 961 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 32.0000000000 | 30.0312500000 | 31.0156250000 | 1 |
| 2 | 31.0156250000 | 30.9843828715 | 31.0000039358 | 5 |
| 3 | 31.0000039358 | 30.9999960642 | 31.0000000000 | all 10 shown |
The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √961 = 31.0000000000 to every decimal shown.
√961 in geometry and everyday measurements
- 961 square feet is 89.3 m². Laid out as a square — a small house footprint or a lot — it is exactly 31 ft (31 ft) on a side.
- 961 is not a sum of two whole-number squares — no pair of squares adds up to it — so √961 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 5 × 6 × 30 box, because 5² + 6² + 30² = 961.
Square roots near √961 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √958 | √958 | 30.9516 | No |
| √959 | √959 | 30.9677 | No |
| √960 | 8√15 | 30.9839 | No |
| √961 | 31 | 31.0000 | Yes |
| √962 | √962 | 31.0161 | No |
| √963 | 3√107 | 31.0322 | No |
| √964 | 2√241 | 31.0483 | No |
- The cube root of 961 is about 9.868272.
- Taking the square root twice gives the fourth root: √31 ≈ 5.567764.
- Squaring undoes the root: (√961)² = 961, while 961² = 923,521 — the number whose square root is 961.
Frequently asked questions
What is the square root of 961?
The square root of 961 is 31, because 31 × 31 = 961. Strictly, 961 has two square roots, 31 and −31; the √ symbol means the positive one.
Is the square root of 961 rational or irrational?
Rational. 961 is a perfect square, so its square root is the whole number 31.
Can √961 be simplified?
Yes, all the way to a whole number: √961 = 31.
What is √961 rounded to two decimal places?
√961 is exactly 31, or 31.00 written to two decimal places.