√958 at a glance
- Exact value
- √958
- Decimal (10 places)
- 30.9515750811
- Rounded
- 31.0 · 30.95 · 30.952
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.951575
- Prime factorization
- 2 × 479
- Cube root
- 9.857993
How to simplify √958
The prime factorization of 958 is 2 × 479. Every prime appears only once, so there is no pair to bring outside the radical — √958 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 958, 2 and 479 appear an odd number of times, so √958 is irrational and 30.9515750811 is a rounded value.
Where √958 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √958 lies between 30 and 31. 958 is 58 above 900 and 3 below 961, so the root is closer to 31.
- Straight line between 900 and 961: 30.9508 (0% low)
- Tangent from 30, i.e. 30 + 58 ÷ 60: 30.9667 (0.05% high)
- Tangent from 31, i.e. 31 − 3 ÷ 62: 30.9516 (0% high)
For √958 the tangent at 31 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 958 is just 3 below 961.
Finding √958 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 958 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 30.9032258065 | 30.9516129032 | 4 |
| 2 | 30.9516129032 | 30.9515372590 | 30.9515750811 | all 10 shown |
Because the starting guess was already close, two steps are enough to match √958 = 30.9515750811 to every decimal shown.
√958 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √958 the pattern is [30; 1, 19, 1, 1, 1, 6, 4, 1, 1, 1, 1, 2, …] with the block of 36 terms after the semicolon repeating forever (only the first 12 of the 36 are shown). A pattern that never ends is one more proof that √958 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 | 9.5 × 10⁻¹ |
| 31/1 | 31.0000000000 | 4.8 × 10⁻² |
| 619/20 | 30.9500000000 | 1.6 × 10⁻³ |
| 650/21 | 30.9523809524 | 8.1 × 10⁻⁴ |
| 1,269/41 | 30.9512195122 | 3.6 × 10⁻⁴ |
| 1,919/62 | 30.9516129032 | 3.8 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 958y² = 1. Its smallest solution in positive whole numbers is x = 16,762,522,330,425,599, y = 541,572,514,048,560 — 17 digits for x, even though 958 is small, which is what makes Pell’s equation famous.
√958 in geometry and everyday measurements
- 958 square feet is 89 m². Laid out as a square — a small house footprint or a lot — it is about 30.95 ft (30 ft 11 in) on a side.
- 958 is not a sum of two whole-number squares — the prime factor 479 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √958 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 15 × 27 box, because 2² + 15² + 27² = 958.
Square roots near √958 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √961 | 31 | 31.0000 | Yes |
- The cube root of 958 is about 9.857993.
- Squaring undoes the root: (√958)² = 958, while 958² = 917,764 — the number whose square root is 958.
Frequently asked questions
What is the square root of 958?
The square root of 958 is √958, about 30.9515750811. The negative root, −30.951575, also squares to 958.
Is the square root of 958 rational or irrational?
Irrational. 958 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 √958 be simplified?
No. 958 = 2 × 479 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √958 rounded to two decimal places?
√958 ≈ 30.95 to two decimal places (31.0 to one, 30.952 to three). Check: 30.95² = 957.9025, close to 958.