√997 at a glance
- Exact value
- √997
- Decimal (10 places)
- 31.5753068077
- Rounded
- 31.6 · 31.58 · 31.575
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.575307
- Prime factorization
- 997
- Cube root
- 9.989990
How to simplify √997
997 is a prime number, so its only factors are 1 and 997. There is no perfect-square factor to pull out, which means √997 is already in its simplest radical form.
The square root of any prime is irrational. If √997 were a fraction a/b in lowest terms, then a² = 997b², so 997 would divide a — and then 997 would divide b too, contradicting “lowest terms.” That is why the decimal 31.5753068077 is only a rounded value.
Where √997 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √997 lies between 31 and 32. 997 is 36 above 961 and 27 below 1,024, so the root is closer to 32.
- Straight line between 961 and 1,024: 31.5714 (0.01% low)
- Tangent from 31, i.e. 31 + 36 ÷ 62: 31.5806 (0.02% high)
- Tangent from 32, i.e. 32 − 27 ÷ 64: 31.5781 (0.01% high)
For √997 the tangent at 32 wins, missing by only 0.0028. Tangent estimates shine when the number sits close to a perfect square — here 997 is just 27 below 1,024.
Finding √997 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 997: following the tangent line down to zero simplifies to averaging x with 997 ÷ x.
Start from the nearest whole number, 32 (32² = 1,024):
| Step | Guess x | 997 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 32.0000000000 | 31.1562500000 | 31.5781250000 | 2 |
| 2 | 31.5781250000 | 31.5724888669 | 31.5753069334 | 6 |
| 3 | 31.5753069334 | 31.5753066819 | 31.5753068077 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √997 = 31.5753068077 to every decimal shown.
√997 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √997 the pattern is [31; 1, 1, 2, 1, 4, 1, 1, 4, 1, 2, 1, 1, …] with the block of 13 terms after the semicolon repeating forever (only the first 12 of the 13 are shown). A pattern that never ends is one more proof that √997 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 | 5.8 × 10⁻¹ |
| 32/1 | 32.0000000000 | 4.2 × 10⁻¹ |
| 63/2 | 31.5000000000 | 7.5 × 10⁻² |
| 158/5 | 31.6000000000 | 2.5 × 10⁻² |
| 221/7 | 31.5714285714 | 3.9 × 10⁻³ |
| 1,042/33 | 31.5757575758 | 4.5 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 997y² = 1. Its smallest solution in positive whole numbers is x = 14,418,057,673, y = 456,624,468. Because the period is odd, the equation with −1 on the right also has a solution: 84,906² − 997 × 2,689² = −1.
√997 in geometry and everyday measurements
- 997 square feet is 92.6 m². Laid out as a square — a small house footprint or a lot — it is about 31.58 ft (31 ft 7 in) on a side.
- 997 = 6² + 31², so by the Pythagorean theorem √997 is the diagonal of a 6 × 31 rectangle — and the distance between the points (0, 0) and (6, 31) on a grid.
Square roots near √997 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √994 | √994 | 31.5278 | No |
| √995 | √995 | 31.5436 | No |
| √996 | 2√249 | 31.5595 | No |
| √997 | √997 | 31.5753 | No |
| √998 | √998 | 31.5911 | No |
| √999 | 3√111 | 31.6070 | No |
| √1000 | 10√10 | 31.6228 | No |
- The cube root of 997 is about 9.989990.
- Squaring undoes the root: (√997)² = 997, while 997² = 994,009 — the number whose square root is 997.
Frequently asked questions
What is the square root of 997?
The square root of 997 is √997, about 31.5753068077. The negative root, −31.575307, also squares to 997.
Is the square root of 997 rational or irrational?
Irrational. 997 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 √997 be simplified?
No. 997 is prime, so there is no perfect square to take out of the radical.
What is √997 rounded to two decimal places?
√997 ≈ 31.58 to two decimal places (31.6 to one, 31.575 to three). Check: 31.58² = 997.2964, close to 997.