√827 at a glance
- Exact value
- √827
- Decimal (10 places)
- 28.7576076891
- Rounded
- 28.8 · 28.76 · 28.758
- Perfect square?
- No — between 28² and 29²
- Rational?
- Irrational
- Both square roots
- ±28.757608
- Prime factorization
- 827
- Cube root
- 9.386460
How to simplify √827
827 is a prime number, so its only factors are 1 and 827. There is no perfect-square factor to pull out, which means √827 is already in its simplest radical form.
The square root of any prime is irrational. If √827 were a fraction a/b in lowest terms, then a² = 827b², so 827 would divide a — and then 827 would divide b too, contradicting “lowest terms.” That is why the decimal 28.7576076891 is only a rounded value.
Where √827 sits between perfect squares
784 = 28² and 841 = 29² are the nearest perfect squares, so √827 lies between 28 and 29. 827 is 43 above 784 and 14 below 841, so the root is closer to 29.
- Straight line between 784 and 841: 28.7544 (0.01% low)
- Tangent from 28, i.e. 28 + 43 ÷ 56: 28.7679 (0.04% high)
- Tangent from 29, i.e. 29 − 14 ÷ 58: 28.7586 (0% high)
For √827 the tangent at 29 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 827 is just 14 below 841.
Finding √827 with the Babylonian method
If a guess is too big, 827 ÷ 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√827) in one step.
Start from the nearest whole number, 29 (29² = 841):
| Step | Guess x | 827 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 29.0000000000 | 28.5172413793 | 28.7586206897 | 2 |
| 2 | 28.7586206897 | 28.7565947242 | 28.7576077069 | 7 |
| 3 | 28.7576077069 | 28.7576076713 | 28.7576076891 | 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 √827 = 28.7576076891 to every decimal shown.
√827 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √827 the pattern is [28; 1, 3, 7, 1, 27, 1, 7, 3, 1, 56] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √827 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 |
|---|---|---|
| 28/1 | 28.0000000000 | 7.6 × 10⁻¹ |
| 29/1 | 29.0000000000 | 2.4 × 10⁻¹ |
| 115/4 | 28.7500000000 | 7.6 × 10⁻³ |
| 834/29 | 28.7586206897 | 1.0 × 10⁻³ |
| 949/33 | 28.7575757576 | 3.2 × 10⁻⁵ |
| 26,457/920 | 28.7576086957 | 1.0 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 827y² = 1. Its smallest solution in positive whole numbers is x = 900,602, y = 31,317.
√827 in geometry and everyday measurements
- 827 square feet is 76.8 m². Laid out as a square — a small house footprint or a lot — it is about 28.76 ft (28 ft 9 in) on a side.
- 827 is not a sum of two whole-number squares — 827 is itself a prime that is one less than a multiple of 4, which rules that out — so √827 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 17 × 23 box, because 3² + 17² + 23² = 827.
Square roots near √827 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √824 | 2√206 | 28.7054 | No |
| √825 | 5√33 | 28.7228 | No |
| √826 | √826 | 28.7402 | No |
| √827 | √827 | 28.7576 | No |
| √828 | 6√23 | 28.7750 | No |
| √829 | √829 | 28.7924 | No |
| √830 | √830 | 28.8097 | No |
- The cube root of 827 is about 9.386460.
- Squaring undoes the root: (√827)² = 827, while 827² = 683,929 — the number whose square root is 827.
Frequently asked questions
What is the square root of 827?
The square root of 827 is √827, about 28.7576076891. The negative root, −28.757608, also squares to 827.
Is the square root of 827 rational or irrational?
Irrational. 827 is not a perfect square — it falls between 784 and 841 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √827 be simplified?
No. 827 is prime, so there is no perfect square to take out of the radical.
What is √827 rounded to two decimal places?
√827 ≈ 28.76 to two decimal places (28.8 to one, 28.758 to three). Check: 28.76² = 827.1376, close to 827.