√970 at a glance
- Exact value
- √970
- Decimal (10 places)
- 31.1448230048
- Rounded
- 31.1 · 31.14 · 31.145
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.144823
- Prime factorization
- 2 × 5 × 97
- Cube root
- 9.898983
How to simplify √970
The prime factorization of 970 is 2 × 5 × 97. Every prime appears only once, so there is no pair to bring outside the radical — √970 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 970, 2, 5 and 97 appear an odd number of times, so √970 is irrational and 31.1448230048 is a rounded value.
Where √970 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √970 lies between 31 and 32. 970 is 9 above 961 and 54 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.1429 (0.01% low)
- Tangent from 31, i.e. 31 + 9 ÷ 62: 31.1452 (0% high)
- Tangent from 32, i.e. 32 − 54 ÷ 64: 31.1563 (0.04% high)
For √970 the tangent at 31 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 970 is just 9 above 961.
Finding √970 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 | 970 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.2903225806 | 31.1451612903 | 3 |
| 2 | 31.1451612903 | 31.1444847229 | 31.1448230066 | 8 |
| 3 | 31.1448230066 | 31.1448230030 | 31.1448230048 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √970 = 31.1448230048 to every decimal shown.
√970 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √970 the pattern is [31; 6, 1, 9, 1, 1, 9, 1, 6, 62] with the block of 9 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √970 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 |
|---|---|---|
| 31/1 | 31.0000000000 | 1.4 × 10⁻¹ |
| 187/6 | 31.1666666667 | 2.2 × 10⁻² |
| 218/7 | 31.1428571429 | 2.0 × 10⁻³ |
| 2,149/69 | 31.1449275362 | 1.0 × 10⁻⁴ |
| 2,367/76 | 31.1447368421 | 8.6 × 10⁻⁵ |
| 4,516/145 | 31.1448275862 | 4.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 970y² = 1. Its smallest solution in positive whole numbers is x = 215,395,035,859, y = 6,915,917,802. Because the period is odd, the equation with −1 on the right also has a solution: 328,173² − 970 × 10,537² = −1.
√970 in geometry and everyday measurements
- 970 square feet is 90.1 m². Laid out as a square — a small house footprint or a lot — it is about 31.14 ft (31 ft 2 in) on a side.
- 970 = 3² + 31² = 21² + 23², so by the Pythagorean theorem √970 is the diagonal of rectangles measuring 3 × 31 and 21 × 23 — and the distance between the points (0, 0) and (3, 31) on a grid.
Square roots near √970 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √967 | √967 | 31.0966 | No |
| √968 | 22√2 | 31.1127 | No |
| √969 | √969 | 31.1288 | No |
| √970 | √970 | 31.1448 | No |
| √971 | √971 | 31.1609 | No |
| √972 | 18√3 | 31.1769 | No |
| √973 | √973 | 31.1929 | No |
- The cube root of 970 is about 9.898983.
- Squaring undoes the root: (√970)² = 970, while 970² = 940,900 — the number whose square root is 970.
Frequently asked questions
What is the square root of 970?
The square root of 970 is √970, about 31.1448230048. The negative root, −31.144823, also squares to 970.
Is the square root of 970 rational or irrational?
Irrational. 970 is not a perfect square — it falls between 961 and 1024 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √970 be simplified?
No. 970 = 2 × 5 × 97 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √970 rounded to two decimal places?
√970 ≈ 31.14 to two decimal places (31.1 to one, 31.145 to three). Check: 31.14² = 969.6996, close to 970.