√957 at a glance
- Exact value
- √957
- Decimal (10 places)
- 30.9354165965
- Rounded
- 30.9 · 30.94 · 30.935
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.935417
- Prime factorization
- 3 × 11 × 29
- Cube root
- 9.854562
How to simplify √957
The prime factorization of 957 is 3 × 11 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √957 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 957, 3, 11 and 29 appear an odd number of times, so √957 is irrational and 30.9354165965 is a rounded value.
Where √957 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √957 lies between 30 and 31. 957 is 57 above 900 and 4 below 961, so the root is closer to 31.
- Straight line between 900 and 961: 30.9344 (0% low)
- Tangent from 30, i.e. 30 + 57 ÷ 60: 30.9500 (0.05% high)
- Tangent from 31, i.e. 31 − 4 ÷ 62: 30.9355 (0% high)
For √957 the tangent at 31 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 957 is just 4 below 961.
Finding √957 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 957: following the tangent line down to zero simplifies to averaging x with 957 ÷ x.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 957 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 30.8709677419 | 30.9354838710 | 4 |
| 2 | 30.9354838710 | 30.9353493222 | 30.9354165966 | 10 |
| 3 | 30.9354165966 | 30.9354165964 | 30.9354165965 | all 10 shown |
The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √957 = 30.9354165965 to every decimal shown.
√957 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √957 the pattern is [30; 1, 14, 2, 14, 1, 60] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √957 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 30/1 | 30.0000000000 | 9.4 × 10⁻¹ |
| 31/1 | 31.0000000000 | 6.5 × 10⁻² |
| 464/15 | 30.9333333333 | 2.1 × 10⁻³ |
| 959/31 | 30.9354838710 | 6.7 × 10⁻⁵ |
| 13,890/449 | 30.9354120267 | 4.6 × 10⁻⁶ |
| 14,849/480 | 30.9354166667 | 7.0 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 957y² = 1. Its smallest solution in positive whole numbers is x = 14,849, y = 480.
√957 in geometry and everyday measurements
- 957 square feet is 88.9 m². Laid out as a square — a small house footprint or a lot — it is about 30.94 ft (30 ft 11 in) on a side.
- 957 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √957 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 13 × 28 box, because 2² + 13² + 28² = 957.
Square roots near √957 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √954 | 3√106 | 30.8869 | No |
| √955 | √955 | 30.9031 | No |
| √956 | 2√239 | 30.9192 | No |
| √957 | √957 | 30.9354 | No |
| √958 | √958 | 30.9516 | No |
| √959 | √959 | 30.9677 | No |
| √960 | 8√15 | 30.9839 | No |
- The cube root of 957 is about 9.854562.
- Squaring undoes the root: (√957)² = 957, while 957² = 915,849 — the number whose square root is 957.
Frequently asked questions
What is the square root of 957?
The square root of 957 is √957, about 30.9354165965. The negative root, −30.935417, also squares to 957.
Is the square root of 957 rational or irrational?
Irrational. 957 is not a perfect square — it falls between 900 and 961 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √957 be simplified?
No. 957 = 3 × 11 × 29 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √957 rounded to two decimal places?
√957 ≈ 30.94 to two decimal places (30.9 to one, 30.935 to three). Check: 30.94² = 957.2836, close to 957.