√980 at a glance
- Exact value
- 14√5
- Decimal (10 places)
- 31.3049516850
- Rounded
- 31.3 · 31.30 · 31.305
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.304952
- Prime factorization
- 2² × 5 × 7²
- Cube root
- 9.932884
How to simplify √980
Look for the largest perfect square that divides 980. Here it is 196 (14²), because 980 = 196 × 5 and 5 has no square factor left:
The prime factorization tells the same story: 980 = 2² × 5 × 7². Each pair of equal primes leaves the radical as one factor, so 2 × 7 comes out and 5 stays inside.
980 has 3 square factors (4, 49 and 196). Starting with a smaller one still works but takes more rounds: √980 = 2√245, and √245 can be simplified again. Using 196 straight away finishes in one step.
Check: (14√5)² = 14² × 5 = 196 × 5 = 980. As a decimal, 14√5 = 14 × 2.2360679775 ≈ 31.3049516850.
Where √980 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √980 lies between 31 and 32. 980 is 19 above 961 and 44 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.3016 (0.01% low)
- Tangent from 31, i.e. 31 + 19 ÷ 62: 31.3065 (0% high)
- Tangent from 32, i.e. 32 − 44 ÷ 64: 31.3125 (0.02% high)
For √980 the tangent at 31 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 980 is just 19 above 961.
Finding √980 with the Babylonian method
Picture a rectangle with an area of 980 and one side x; the other side must be 980 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √980.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 980 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.6129032258 | 31.3064516129 | 2 |
| 2 | 31.3064516129 | 31.3034518290 | 31.3049517209 | 7 |
| 3 | 31.3049517209 | 31.3049516491 | 31.3049516850 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √980 = 31.3049516850 to every decimal shown.
√980 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √980 the pattern is [31; 3, 3, 1, 1, 2, 1, 1, 3, 3, 62] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √980 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 | 3.0 × 10⁻¹ |
| 94/3 | 31.3333333333 | 2.8 × 10⁻² |
| 313/10 | 31.3000000000 | 5.0 × 10⁻³ |
| 407/13 | 31.3076923077 | 2.7 × 10⁻³ |
| 720/23 | 31.3043478261 | 6.0 × 10⁻⁴ |
| 1,847/59 | 31.3050847458 | 1.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 980y² = 1. Its smallest solution in positive whole numbers is x = 51,841, y = 1,656.
√980 in geometry and everyday measurements
- 980 square feet is 91 m². Laid out as a square — a small house footprint or a lot — it is about 31.3 ft (31 ft 4 in) on a side.
- 980 = 14² + 28², so by the Pythagorean theorem √980 is the diagonal of a 14 × 28 rectangle — and the distance between the points (0, 0) and (14, 28) on a grid.
- Since √980 = 14√5, a length of √980 is exactly 14 copies of the length √5 laid end to end.
Square roots near √980 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √977 | √977 | 31.2570 | No |
| √978 | √978 | 31.2730 | No |
| √979 | √979 | 31.2890 | No |
| √980 | 14√5 | 31.3050 | No |
| √981 | 3√109 | 31.3209 | No |
| √982 | √982 | 31.3369 | No |
| √983 | √983 | 31.3528 | No |
- The cube root of 980 is about 9.932884.
- Because 980 = 4 × 245, the root is twice √245: 2 × 15.652476 ≈ 31.304952.
Frequently asked questions
What is the square root of 980?
The square root of 980 is 14√5 in simplest radical form, which is about 31.3049516850. The negative root, −31.304952, also squares to 980.
Is the square root of 980 rational or irrational?
Irrational. 980 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 √980 be simplified?
Yes. The largest perfect square dividing 980 is 196, so √980 = √196 × √5 = 14√5.
What is √980 rounded to two decimal places?
√980 ≈ 31.30 to two decimal places (31.3 to one, 31.305 to three). Check: 31.30² = 979.69, close to 980.