√990 at a glance
- Exact value
- 3√110
- Decimal (10 places)
- 31.4642654451
- Rounded
- 31.5 · 31.46 · 31.464
- Perfect square?
- No — between 31² and 32²
- Rational?
- Irrational
- Both square roots
- ±31.464265
- Prime factorization
- 2 × 3² × 5 × 11
- Cube root
- 9.966555
How to simplify √990
Look for the largest perfect square that divides 990. Here it is 9 (3²), because 990 = 9 × 110 and 110 has no square factor left:
The prime factorization tells the same story: 990 = 2 × 3² × 5 × 11. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 5 × 11 stays inside.
Check: (3√110)² = 3² × 110 = 9 × 110 = 990. As a decimal, 3√110 = 3 × 10.4880884817 ≈ 31.4642654451.
Where √990 sits between perfect squares
961 = 31² and 1,024 = 32² are the nearest perfect squares, so √990 lies between 31 and 32. 990 is 29 above 961 and 34 below 1,024, so the root is closer to 31.
- Straight line between 961 and 1,024: 31.4603 (0.01% low)
- Tangent from 31, i.e. 31 + 29 ÷ 62: 31.4677 (0.01% high)
- Tangent from 32, i.e. 32 − 34 ÷ 64: 31.4688 (0.01% high)
For √990 the tangent at 31 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 990 is just 29 above 961.
Finding √990 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 | 990 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 31.9354838710 | 31.4677419355 | 2 |
| 2 | 31.4677419355 | 31.4607893388 | 31.4642656371 | 6 |
| 3 | 31.4642656371 | 31.4642652531 | 31.4642654451 | 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 √990 = 31.4642654451 to every decimal shown.
√990 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √990 the pattern is [31; 2, 6, 2, 62] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √990 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 | 4.6 × 10⁻¹ |
| 63/2 | 31.5000000000 | 3.6 × 10⁻² |
| 409/13 | 31.4615384615 | 2.7 × 10⁻³ |
| 881/28 | 31.4642857143 | 2.0 × 10⁻⁵ |
| 55,031/1,749 | 31.4642652945 | 1.5 × 10⁻⁷ |
| 110,943/3,526 | 31.4642654566 | 1.2 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 990y² = 1. Its smallest solution in positive whole numbers is x = 881, y = 28.
√990 in geometry and everyday measurements
- 990 square feet is 92 m². Laid out as a square — a small house footprint or a lot — it is about 31.46 ft (31 ft 6 in) on a side.
- 990 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √990 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 5 × 31 box, because 2² + 5² + 31² = 990.
- Since √990 = 3√110, a length of √990 is exactly 3 copies of the length √110 laid end to end.
Square roots near √990 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √987 | √987 | 31.4166 | No |
| √988 | 2√247 | 31.4325 | No |
| √989 | √989 | 31.4484 | No |
| √990 | 3√110 | 31.4643 | No |
| √991 | √991 | 31.4802 | No |
| √992 | 4√62 | 31.4960 | No |
| √993 | √993 | 31.5119 | No |
- The cube root of 990 is about 9.966555.
- Squaring undoes the root: (√990)² = 990, while 990² = 980,100 — the number whose square root is 990.
Frequently asked questions
What is the square root of 990?
The square root of 990 is 3√110 in simplest radical form, which is about 31.4642654451. The negative root, −31.464265, also squares to 990.
Is the square root of 990 rational or irrational?
Irrational. 990 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 √990 be simplified?
Yes. The largest perfect square dividing 990 is 9, so √990 = √9 × √110 = 3√110.
What is √990 rounded to two decimal places?
√990 ≈ 31.46 to two decimal places (31.5 to one, 31.464 to three). Check: 31.46² = 989.7316, close to 990.