√887 at a glance
- Exact value
- √887
- Decimal (10 places)
- 29.7825452237
- Rounded
- 29.8 · 29.78 · 29.783
- Perfect square?
- No — between 29² and 30²
- Rational?
- Irrational
- Both square roots
- ±29.782545
- Prime factorization
- 887
- Cube root
- 9.608182
How to simplify √887
887 is a prime number, so its only factors are 1 and 887. There is no perfect-square factor to pull out, which means √887 is already in its simplest radical form.
The square root of any prime is irrational. If √887 were a fraction a/b in lowest terms, then a² = 887b², so 887 would divide a — and then 887 would divide b too, contradicting “lowest terms.” That is why the decimal 29.7825452237 is only a rounded value.
Where √887 sits between perfect squares
841 = 29² and 900 = 30² are the nearest perfect squares, so √887 lies between 29 and 30. 887 is 46 above 841 and 13 below 900, so the root is closer to 30.
- Straight line between 841 and 900: 29.7797 (0.01% low)
- Tangent from 29, i.e. 29 + 46 ÷ 58: 29.7931 (0.04% high)
- Tangent from 30, i.e. 30 − 13 ÷ 60: 29.7833 (0% high)
For √887 the tangent at 30 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 887 is just 13 below 900.
Finding √887 with the Babylonian method
If a guess is too big, 887 ÷ 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√887) in one step.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 887 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 29.5666666667 | 29.7833333333 | 3 |
| 2 | 29.7833333333 | 29.7817571349 | 29.7825452341 | 7 |
| 3 | 29.7825452341 | 29.7825452132 | 29.7825452237 | 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 √887 = 29.7825452237 to every decimal shown.
√887 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √887 the pattern is [29; 1, 3, 1, 1, 2, 29, 2, 1, 1, 3, 1, 58] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √887 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 |
|---|---|---|
| 29/1 | 29.0000000000 | 7.8 × 10⁻¹ |
| 30/1 | 30.0000000000 | 2.2 × 10⁻¹ |
| 119/4 | 29.7500000000 | 3.3 × 10⁻² |
| 149/5 | 29.8000000000 | 1.7 × 10⁻² |
| 268/9 | 29.7777777778 | 4.8 × 10⁻³ |
| 685/23 | 29.7826086957 | 6.3 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 887y² = 1. Its smallest solution in positive whole numbers is x = 469,224, y = 15,755.
√887 in geometry and everyday measurements
- 887 square feet is 82.4 m². Laid out as a square — a small house footprint or a lot — it is about 29.78 ft (29 ft 9 in) on a side.
- 887 is not a sum of two whole-number squares — 887 is itself a prime that is one less than a multiple of 4, which rules that out — so √887 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √887 as its space diagonal.
Square roots near √887 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √884 | 2√221 | 29.7321 | No |
| √885 | √885 | 29.7489 | No |
| √886 | √886 | 29.7658 | No |
| √887 | √887 | 29.7825 | No |
| √888 | 2√222 | 29.7993 | No |
| √889 | √889 | 29.8161 | No |
| √890 | √890 | 29.8329 | No |
- The cube root of 887 is about 9.608182.
- Squaring undoes the root: (√887)² = 887, while 887² = 786,769 — the number whose square root is 887.
Frequently asked questions
What is the square root of 887?
The square root of 887 is √887, about 29.7825452237. The negative root, −29.782545, also squares to 887.
Is the square root of 887 rational or irrational?
Irrational. 887 is not a perfect square — it falls between 841 and 900 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √887 be simplified?
No. 887 is prime, so there is no perfect square to take out of the radical.
What is √887 rounded to two decimal places?
√887 ≈ 29.78 to two decimal places (29.8 to one, 29.783 to three). Check: 29.78² = 886.8484, close to 887.