√974 at a glance
- Exact value
- √974
- Decimal (10 places)
- 31.2089730687
- Rounded
- 31.2 · 31.21 · 31.209
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.208973
- Prime factorization
- 2 × 487
- Cube root
- 9.912571
How to simplify √974
The prime factorization of 974 is 2 × 487. Every prime appears only once, so there is no pair to bring outside the radical — √974 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 974, 2 and 487 appear an odd number of times, so √974 is irrational and 31.2089730687 is a rounded value.
Where √974 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √974 lies between 31 and 32. 974 is 13 above 961 and 50 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.2063 (0.01% low)
- Tangent from 31, i.e. 31 + 13 ÷ 62: 31.2097 (0% high)
- Tangent from 32, i.e. 32 − 50 ÷ 64: 31.2188 (0.03% high)
For √974 the tangent at 31 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 974 is just 13 above 961.
Finding √974 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 | 974 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.4193548387 | 31.2096774194 | 3 |
| 2 | 31.2096774194 | 31.2082687339 | 31.2089730766 | 8 |
| 3 | 31.2089730766 | 31.2089730607 | 31.2089730687 | 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 √974 = 31.2089730687 to every decimal shown.
√974 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √974 the pattern is [31; 4, 1, 3, 1, 1, 1, 11, 1, 5, 3, 8, 1, …] with the block of 32 terms after the semicolon repeating forever (only the first 12 of the 32 are shown). A pattern that never ends is one more proof that √974 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 | 2.1 × 10⁻¹ |
| 125/4 | 31.2500000000 | 4.1 × 10⁻² |
| 156/5 | 31.2000000000 | 9.0 × 10⁻³ |
| 593/19 | 31.2105263158 | 1.6 × 10⁻³ |
| 749/24 | 31.2083333333 | 6.4 × 10⁻⁴ |
| 1,342/43 | 31.2093023256 | 3.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 974y² = 1. Its smallest solution in positive whole numbers is x = 488,825,745,235,215, y = 15,662,987,185,124 — 15 digits for x, even though 974 is small, which is what makes Pell’s equation famous.
√974 in geometry and everyday measurements
- 974 square feet is 90.5 m². Laid out as a square — a small house footprint or a lot — it is about 31.21 ft (31 ft 3 in) on a side.
- 974 is not a sum of two whole-number squares — the prime factor 487 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √974 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 31 box, because 2² + 3² + 31² = 974.
Square roots near √974 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √971 | √971 | 31.1609 | No |
| √972 | 18√3 | 31.1769 | No |
| √973 | √973 | 31.1929 | No |
| √974 | √974 | 31.2090 | No |
| √975 | 5√39 | 31.2250 | No |
| √976 | 4√61 | 31.2410 | No |
| √977 | √977 | 31.2570 | No |
- The cube root of 974 is about 9.912571.
- Squaring undoes the root: (√974)² = 974, while 974² = 948,676 — the number whose square root is 974.
Frequently asked questions
What is the square root of 974?
The square root of 974 is √974, about 31.2089730687. The negative root, −31.208973, also squares to 974.
Is the square root of 974 rational or irrational?
Irrational. 974 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 √974 be simplified?
No. 974 = 2 × 487 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √974 rounded to two decimal places?
√974 ≈ 31.21 to two decimal places (31.2 to one, 31.209 to three). Check: 31.21² = 974.0641, close to 974.