√973 at a glance
- Exact value
- √973
- Decimal (10 places)
- 31.1929479210
- Rounded
- 31.2 · 31.19 · 31.193
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.192948
- Prime factorization
- 7 × 139
- Cube root
- 9.909178
How to simplify √973
The prime factorization of 973 is 7 × 139. Every prime appears only once, so there is no pair to bring outside the radical — √973 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 973, 7 and 139 appear an odd number of times, so √973 is irrational and 31.1929479210 is a rounded value.
Where √973 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √973 lies between 31 and 32. 973 is 12 above 961 and 51 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.1905 (0.01% low)
- Tangent from 31, i.e. 31 + 12 ÷ 62: 31.1935 (0% high)
- Tangent from 32, i.e. 32 − 51 ÷ 64: 31.2031 (0.03% high)
For √973 the tangent at 31 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 973 is just 12 above 961.
Finding √973 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 973: following the tangent line down to zero simplifies to averaging x with 973 ÷ x.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 973 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.3870967742 | 31.1935483871 | 3 |
| 2 | 31.1935483871 | 31.1923474664 | 31.1929479267 | 8 |
| 3 | 31.1929479267 | 31.1929479152 | 31.1929479210 | 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 √973 = 31.1929479210 to every decimal shown.
√973 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √973 the pattern is [31; 5, 5, 2, 8, 2, 5, 5, 62] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √973 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 | 1.9 × 10⁻¹ |
| 156/5 | 31.2000000000 | 7.1 × 10⁻³ |
| 811/26 | 31.1923076923 | 6.4 × 10⁻⁴ |
| 1,778/57 | 31.1929824561 | 3.5 × 10⁻⁵ |
| 15,035/482 | 31.1929460581 | 1.9 × 10⁻⁶ |
| 31,848/1,021 | 31.1929480901 | 1.7 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 973y² = 1. Its smallest solution in positive whole numbers is x = 903,223, y = 28,956.
√973 in geometry and everyday measurements
- 973 square feet is 90.4 m². Laid out as a square — a small house footprint or a lot — it is about 31.19 ft (31 ft 2 in) on a side.
- 973 is not a sum of two whole-number squares — the prime factor 7 and 139 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √973 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 8 × 30 box, because 3² + 8² + 30² = 973.
Square roots near √973 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √975 | 5√39 | 31.2250 | No |
| √976 | 4√61 | 31.2410 | No |
- The cube root of 973 is about 9.909178.
- Squaring undoes the root: (√973)² = 973, while 973² = 946,729 — the number whose square root is 973.
Frequently asked questions
What is the square root of 973?
The square root of 973 is √973, about 31.1929479210. The negative root, −31.192948, also squares to 973.
Is the square root of 973 rational or irrational?
Irrational. 973 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 √973 be simplified?
No. 973 = 7 × 139 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √973 rounded to two decimal places?
√973 ≈ 31.19 to two decimal places (31.2 to one, 31.193 to three). Check: 31.19² = 972.8161, close to 973.