√707 at a glance
- Exact value
- √707
- Decimal (10 places)
- 26.5894716006
- Rounded
- 26.6 · 26.59 · 26.589
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.589472
- Prime factorization
- 7 × 101
- Cube root
- 8.908539
How to simplify √707
The prime factorization of 707 is 7 × 101. Every prime appears only once, so there is no pair to bring outside the radical — √707 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 707, 7 and 101 appear an odd number of times, so √707 is irrational and 26.5894716006 is a rounded value.
Where √707 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √707 lies between 26 and 27. 707 is 31 above 676 and 22 below 729, so the root is closer to 27.
- Straight line between 676 and 729: 26.5849 (0.02% low)
- Tangent from 26, i.e. 26 + 31 ÷ 52: 26.5962 (0.03% high)
- Tangent from 27, i.e. 27 − 22 ÷ 54: 26.5926 (0.01% high)
For √707 the tangent at 27 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 707 is just 22 below 729.
Finding √707 with the Babylonian method
If a guess is too big, 707 ÷ 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√707) in one step.
Start from the nearest whole number, 27 (27² = 729):
| Step | Guess x | 707 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 26.1851851852 | 26.5925925926 | 2 |
| 2 | 26.5925925926 | 26.5863509749 | 26.5894717838 | 6 |
| 3 | 26.5894717838 | 26.5894714175 | 26.5894716006 | 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 √707 = 26.5894716006 to every decimal shown.
√707 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √707 the pattern is [26; 1, 1, 2, 3, 2, 1, 1, 52] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √707 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 |
|---|---|---|
| 26/1 | 26.0000000000 | 5.9 × 10⁻¹ |
| 27/1 | 27.0000000000 | 4.1 × 10⁻¹ |
| 53/2 | 26.5000000000 | 8.9 × 10⁻² |
| 133/5 | 26.6000000000 | 1.1 × 10⁻² |
| 452/17 | 26.5882352941 | 1.2 × 10⁻³ |
| 1,037/39 | 26.5897435897 | 2.7 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 707y² = 1. Its smallest solution in positive whole numbers is x = 2,526, y = 95.
√707 in geometry and everyday measurements
- 707 square feet is 65.7 m². Laid out as a square — a small house footprint or a lot — it is about 26.59 ft (26 ft 7 in) on a side.
- 707 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √707 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 25 box, because 1² + 9² + 25² = 707.
Square roots near √707 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √704 | 8√11 | 26.5330 | No |
| √705 | √705 | 26.5518 | No |
| √706 | √706 | 26.5707 | No |
| √707 | √707 | 26.5895 | No |
| √708 | 2√177 | 26.6083 | No |
| √709 | √709 | 26.6271 | No |
| √710 | √710 | 26.6458 | No |
- The cube root of 707 is about 8.908539.
- Squaring undoes the root: (√707)² = 707, while 707² = 499,849 — the number whose square root is 707.
Frequently asked questions
What is the square root of 707?
The square root of 707 is √707, about 26.5894716006. The negative root, −26.589472, also squares to 707.
Is the square root of 707 rational or irrational?
Irrational. 707 is not a perfect square — it falls between 676 and 729 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √707 be simplified?
No. 707 = 7 × 101 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √707 rounded to two decimal places?
√707 ≈ 26.59 to two decimal places (26.6 to one, 26.589 to three). Check: 26.59² = 707.0281, close to 707.