√978 at a glance
- Exact value
- √978
- Decimal (10 places)
- 31.2729915422
- Rounded
- 31.3 · 31.27 · 31.273
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.272992
- Prime factorization
- 2 × 3 × 163
- Cube root
- 9.926122
How to simplify √978
The prime factorization of 978 is 2 × 3 × 163. Every prime appears only once, so there is no pair to bring outside the radical — √978 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 978, 2, 3 and 163 appear an odd number of times, so √978 is irrational and 31.2729915422 is a rounded value.
Where √978 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √978 lies between 31 and 32. 978 is 17 above 961 and 46 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.2698 (0.01% low)
- Tangent from 31, i.e. 31 + 17 ÷ 62: 31.2742 (0% high)
- Tangent from 32, i.e. 32 − 46 ÷ 64: 31.2813 (0.03% high)
For √978 the tangent at 31 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 978 is just 17 above 961.
Finding √978 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 978 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.5483870968 | 31.2741935484 | 2 |
| 2 | 31.2741935484 | 31.2717895823 | 31.2729915653 | 7 |
| 3 | 31.2729915653 | 31.2729915191 | 31.2729915422 | 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 √978 = 31.2729915422 to every decimal shown.
√978 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √978 the pattern is [31; 3, 1, 1, 1, 30, 1, 1, 1, 3, 62] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √978 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 | 2.7 × 10⁻¹ |
| 94/3 | 31.3333333333 | 6.0 × 10⁻² |
| 125/4 | 31.2500000000 | 2.3 × 10⁻² |
| 219/7 | 31.2857142857 | 1.3 × 10⁻² |
| 344/11 | 31.2727272727 | 2.6 × 10⁻⁴ |
| 10,539/337 | 31.2729970326 | 5.5 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 978y² = 1. Its smallest solution in positive whole numbers is x = 118,337, y = 3,784.
√978 in geometry and everyday measurements
- 978 square feet is 90.9 m². Laid out as a square — a small house footprint or a lot — it is about 31.27 ft (31 ft 3 in) on a side.
- 978 is not a sum of two whole-number squares — the prime factor 3 and 163 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √978 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 31 box, because 1² + 4² + 31² = 978.
Square roots near √978 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √980 | 14√5 | 31.3050 | No |
| √981 | 3√109 | 31.3209 | No |
- The cube root of 978 is about 9.926122.
- Squaring undoes the root: (√978)² = 978, while 978² = 956,484 — the number whose square root is 978.
Frequently asked questions
What is the square root of 978?
The square root of 978 is √978, about 31.2729915422. The negative root, −31.272992, also squares to 978.
Is the square root of 978 rational or irrational?
Irrational. 978 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 √978 be simplified?
No. 978 = 2 × 3 × 163 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √978 rounded to two decimal places?
√978 ≈ 31.27 to two decimal places (31.3 to one, 31.273 to three). Check: 31.27² = 977.8129, close to 978.