√971 at a glance
- Exact value
- √971
- Decimal (10 places)
- 31.1608729018
- Rounded
- 31.2 · 31.16 · 31.161
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.160873
- Prime factorization
- 971
- Cube root
- 9.902384
How to simplify √971
971 is a prime number, so its only factors are 1 and 971. There is no perfect-square factor to pull out, which means √971 is already in its simplest radical form.
The square root of any prime is irrational. If √971 were a fraction a/b in lowest terms, then a² = 971b², so 971 would divide a — and then 971 would divide b too, contradicting “lowest terms.” That is why the decimal 31.1608729018 is only a rounded value.
Where √971 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √971 lies between 31 and 32. 971 is 10 above 961 and 53 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.1587 (0.01% low)
- Tangent from 31, i.e. 31 + 10 ÷ 62: 31.1613 (0% high)
- Tangent from 32, i.e. 32 − 53 ÷ 64: 31.1719 (0.04% high)
For √971 the tangent at 31 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 971 is just 10 above 961.
Finding √971 with the Babylonian method
If a guess is too big, 971 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√971) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 971 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.3225806452 | 31.1612903226 | 3 |
| 2 | 31.1612903226 | 31.1604554865 | 31.1608729046 | 8 |
| 3 | 31.1608729046 | 31.1608728990 | 31.1608729018 | 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 √971 = 31.1608729018 to every decimal shown.
√971 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √971 the pattern is [31; 6, 4, 1, 1, 1, 2, 5, 3, 2, 12, 31, 12, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √971 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 | 1.6 × 10⁻¹ |
| 187/6 | 31.1666666667 | 5.8 × 10⁻³ |
| 779/25 | 31.1600000000 | 8.7 × 10⁻⁴ |
| 966/31 | 31.1612903226 | 4.2 × 10⁻⁴ |
| 1,745/56 | 31.1607142857 | 1.6 × 10⁻⁴ |
| 2,711/87 | 31.1609195402 | 4.7 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 971y² = 1. Its smallest solution in positive whole numbers is x = 12,479,806,786,330, y = 400,496,058,813 — 14 digits for x, even though 971 is small, which is what makes Pell’s equation famous.
√971 in geometry and everyday measurements
- 971 square feet is 90.2 m². Laid out as a square — a small house footprint or a lot — it is about 31.16 ft (31 ft 2 in) on a side.
- 971 is not a sum of two whole-number squares — 971 is itself a prime that is one less than a multiple of 4, which rules that out — so √971 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 31 box, because 1² + 3² + 31² = 971.
Square roots near √971 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √974 | √974 | 31.2090 | No |
- The cube root of 971 is about 9.902384.
- Squaring undoes the root: (√971)² = 971, while 971² = 942,841 — the number whose square root is 971.
Frequently asked questions
What is the square root of 971?
The square root of 971 is √971, about 31.1608729018. The negative root, −31.160873, also squares to 971.
Is the square root of 971 rational or irrational?
Irrational. 971 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 √971 be simplified?
No. 971 is prime, so there is no perfect square to take out of the radical.
What is √971 rounded to two decimal places?
√971 ≈ 31.16 to two decimal places (31.2 to one, 31.161 to three). Check: 31.16² = 970.9456, close to 971.