√722 at a glance
- Exact value
- 19√2
- Decimal (10 places)
- 26.8700576851
- Rounded
- 26.9 · 26.87 · 26.870
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.870058
- Prime factorization
- 2 × 19²
- Cube root
- 8.971101
How to simplify √722
Look for the largest perfect square that divides 722. Here it is 361 (19²), because 722 = 361 × 2 and 2 has no square factor left:
The prime factorization tells the same story: 722 = 2 × 19². Each pair of equal primes leaves the radical as one factor, so 19 comes out and 2 stays inside.
Check: (19√2)² = 19² × 2 = 361 × 2 = 722. As a decimal, 19√2 = 19 × 1.4142135624 ≈ 26.8700576851.
Where √722 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √722 lies between 26 and 27. 722 is 46 above 676 and 7 below 729, so the root is closer to 27.
- Straight line between 676 and 729: 26.8679 (0.01% low)
- Tangent from 26, i.e. 26 + 46 ÷ 52: 26.8846 (0.05% high)
- Tangent from 27, i.e. 27 − 7 ÷ 54: 26.8704 (0% high)
For √722 the tangent at 27 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 722 is just 7 below 729.
Finding √722 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, 27 (27² = 729):
| Step | Guess x | 722 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 26.7407407407 | 26.8703703704 | 3 |
| 2 | 26.8703703704 | 26.8697450034 | 26.8700576869 | 8 |
| 3 | 26.8700576869 | 26.8700576833 | 26.8700576851 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √722 = 26.8700576851 to every decimal shown.
√722 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √722 the pattern is [26; 1, 6, 1, 2, 3, 2, 26, 2, 3, 2, 1, 6, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √722 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 | 8.7 × 10⁻¹ |
| 27/1 | 27.0000000000 | 1.3 × 10⁻¹ |
| 188/7 | 26.8571428571 | 1.3 × 10⁻² |
| 215/8 | 26.8750000000 | 4.9 × 10⁻³ |
| 618/23 | 26.8695652174 | 4.9 × 10⁻⁴ |
| 2,069/77 | 26.8701298701 | 7.2 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 722y² = 1. Its smallest solution in positive whole numbers is x = 22,619,537, y = 841,812.
√722 in geometry and everyday measurements
- 722 square feet is 67.1 m². Laid out as a square — a small house footprint or a lot — it is about 26.87 ft (26 ft 10 in) on a side.
- 722 = 19² + 19², so by the Pythagorean theorem √722 is the diagonal of a 19 × 19 rectangle — and the distance between the points (0, 0) and (19, 19) on a grid.
- Since √722 = 19√2, a length of √722 is exactly 19 copies of the length √2 laid end to end.
Square roots near √722 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √719 | √719 | 26.8142 | No |
| √720 | 12√5 | 26.8328 | No |
| √721 | √721 | 26.8514 | No |
| √722 | 19√2 | 26.8701 | No |
| √723 | √723 | 26.8887 | No |
| √724 | 2√181 | 26.9072 | No |
| √725 | 5√29 | 26.9258 | No |
- The cube root of 722 is about 8.971101.
- Squaring undoes the root: (√722)² = 722, while 722² = 521,284 — the number whose square root is 722.
Frequently asked questions
What is the square root of 722?
The square root of 722 is 19√2 in simplest radical form, which is about 26.8700576851. The negative root, −26.870058, also squares to 722.
Is the square root of 722 rational or irrational?
Irrational. 722 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 √722 be simplified?
Yes. The largest perfect square dividing 722 is 361, so √722 = √361 × √2 = 19√2.
What is √722 rounded to two decimal places?
√722 ≈ 26.87 to two decimal places (26.9 to one, 26.870 to three). Check: 26.87² = 721.9969, close to 722.