√987 at a glance
- Exact value
- √987
- Decimal (10 places)
- 31.4165561448
- Rounded
- 31.4 · 31.42 · 31.417
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.416556
- Prime factorization
- 3 × 7 × 47
- Cube root
- 9.956478
How to simplify √987
The prime factorization of 987 is 3 × 7 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √987 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 987, 3, 7 and 47 appear an odd number of times, so √987 is irrational and 31.4165561448 is a rounded value.
Where √987 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √987 lies between 31 and 32. 987 is 26 above 961 and 37 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.4127 (0.01% low)
- Tangent from 31, i.e. 31 + 26 ÷ 62: 31.4194 (0.01% high)
- Tangent from 32, i.e. 32 − 37 ÷ 64: 31.4219 (0.02% high)
For √987 the tangent at 31 wins, missing by only 0.0028. Tangent estimates shine when the number sits close to a perfect square — here 987 is just 26 above 961.
Finding √987 with the Babylonian method
If a guess is too big, 987 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√987) in one step.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 987 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.8387096774 | 31.4193548387 | 2 |
| 2 | 31.4193548387 | 31.4137577002 | 31.4165562695 | 6 |
| 3 | 31.4165562695 | 31.4165560202 | 31.4165561448 | 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 √987 = 31.4165561448 to every decimal shown.
√987 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √987 the pattern is [31; 2, 2, 2, 62] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √987 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 | 4.2 × 10⁻¹ |
| 63/2 | 31.5000000000 | 8.3 × 10⁻² |
| 157/5 | 31.4000000000 | 1.7 × 10⁻² |
| 377/12 | 31.4166666667 | 1.1 × 10⁻⁴ |
| 23,531/749 | 31.4165554072 | 7.4 × 10⁻⁷ |
| 47,439/1,510 | 31.4165562914 | 1.5 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 987y² = 1. Its smallest solution in positive whole numbers is x = 377, y = 12.
√987 in geometry and everyday measurements
- 987 square feet is 91.7 m². Laid out as a square — a small house footprint or a lot — it is about 31.42 ft (31 ft 5 in) on a side.
- 987 is not a sum of two whole-number squares — the prime factor 3, 7 and 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √987 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 31 box, because 1² + 5² + 31² = 987.
Square roots near √987 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √984 | 2√246 | 31.3688 | No |
| √985 | √985 | 31.3847 | No |
| √986 | √986 | 31.4006 | No |
| √987 | √987 | 31.4166 | No |
| √988 | 2√247 | 31.4325 | No |
| √989 | √989 | 31.4484 | No |
| √990 | 3√110 | 31.4643 | No |
- The cube root of 987 is about 9.956478.
- Squaring undoes the root: (√987)² = 987, while 987² = 974,169 — the number whose square root is 987.
Frequently asked questions
What is the square root of 987?
The square root of 987 is √987, about 31.4165561448. The negative root, −31.416556, also squares to 987.
Is the square root of 987 rational or irrational?
Irrational. 987 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 √987 be simplified?
No. 987 = 3 × 7 × 47 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √987 rounded to two decimal places?
√987 ≈ 31.42 to two decimal places (31.4 to one, 31.417 to three). Check: 31.42² = 987.2164, close to 987.