√976 at a glance
- Exact value
- 4√61
- Decimal (10 places)
- 31.2409987036
- Rounded
- 31.2 · 31.24 · 31.241
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.240999
- Prime factorization
- 2⁴ × 61
- Cube root
- 9.919351
How to simplify √976
Look for the largest perfect square that divides 976. Here it is 16 (4²), because 976 = 16 × 61 and 61 has no square factor left:
The prime factorization tells the same story: 976 = 2⁴ × 61. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 61 stays inside.
976 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √976 = 2√244, and √244 can be simplified again. Using 16 straight away finishes in one step.
Check: (4√61)² = 4² × 61 = 16 × 61 = 976. As a decimal, 4√61 = 4 × 7.8102496759 ≈ 31.2409987036.
Where √976 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √976 lies between 31 and 32. 976 is 15 above 961 and 48 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.2381 (0.01% low)
- Tangent from 31, i.e. 31 + 15 ÷ 62: 31.2419 (0% high)
- Tangent from 32, i.e. 32 − 48 ÷ 64: 31.2500 (0.03% high)
For √976 the tangent at 31 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 976 is just 15 above 961.
Finding √976 with the Babylonian method
Picture a rectangle with an area of 976 and one side x; the other side must be 976 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √976.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 976 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.4838709677 | 31.2419354839 | 3 |
| 2 | 31.2419354839 | 31.2400619515 | 31.2409987177 | 7 |
| 3 | 31.2409987177 | 31.2409986896 | 31.2409987036 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √976 = 31.2409987036 to every decimal shown.
√976 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √976 the pattern is [31; 4, 6, 1, 2, 3, 1, 4, 2, 3, 2, 4, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √976 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 | 2.4 × 10⁻¹ |
| 125/4 | 31.2500000000 | 9.0 × 10⁻³ |
| 781/25 | 31.2400000000 | 1.0 × 10⁻³ |
| 906/29 | 31.2413793103 | 3.8 × 10⁻⁴ |
| 2,593/83 | 31.2409638554 | 3.5 × 10⁻⁵ |
| 8,685/278 | 31.2410071942 | 8.5 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 976y² = 1. Its smallest solution in positive whole numbers is x = 1,766,319,049, y = 56,538,495.
√976 in geometry and everyday measurements
- 976 square feet is 90.7 m². Laid out as a square — a small house footprint or a lot — it is about 31.24 ft (31 ft 3 in) on a side.
- 976 = 20² + 24², so by the Pythagorean theorem √976 is the diagonal of a 20 × 24 rectangle — and the distance between the points (0, 0) and (20, 24) on a grid.
- Since √976 = 4√61, a length of √976 is exactly 4 copies of the length √61 laid end to end.
Square roots near √976 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √978 | √978 | 31.2730 | No |
| √979 | √979 | 31.2890 | No |
- The cube root of 976 is about 9.919351.
- Because 976 = 4 × 244, the root is twice √244: 2 × 15.620499 ≈ 31.240999.
Frequently asked questions
What is the square root of 976?
The square root of 976 is 4√61 in simplest radical form, which is about 31.2409987036. The negative root, −31.240999, also squares to 976.
Is the square root of 976 rational or irrational?
Irrational. 976 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 √976 be simplified?
Yes. The largest perfect square dividing 976 is 16, so √976 = √16 × √61 = 4√61.
What is √976 rounded to two decimal places?
√976 ≈ 31.24 to two decimal places (31.2 to one, 31.241 to three). Check: 31.24² = 975.9376, close to 976.