√720 at a glance
- Exact value
- 12√5
- Decimal (10 places)
- 26.8328157300
- Rounded
- 26.8 · 26.83 · 26.833
- Perfect square?
- No — between 26² and 27²
- Rational?
- Irrational
- Both square roots
- ±26.832816
- Prime factorization
- 2⁴ × 3² × 5
- Cube root
- 8.962809
How to simplify √720
Look for the largest perfect square that divides 720. Here it is 144 (12²), because 720 = 144 × 5 and 5 has no square factor left:
The prime factorization tells the same story: 720 = 2⁴ × 3² × 5. Each pair of equal primes leaves the radical as one factor, so 2² × 3 comes out and 5 stays inside.
720 has 5 square factors (4, 9, 16, 36 and 144). Starting with a smaller one still works but takes more rounds: √720 = 2√180, and √180 can be simplified again. Using 144 straight away finishes in one step.
Check: (12√5)² = 12² × 5 = 144 × 5 = 720. As a decimal, 12√5 = 12 × 2.2360679775 ≈ 26.8328157300.
Where √720 sits between perfect squares
676 = 26² and 729 = 27² are the nearest perfect squares, so √720 lies between 26 and 27. 720 is 44 above 676 and 9 below 729, so the root is closer to 27.
- Straight line between 676 and 729: 26.8302 (0.01% low)
- Tangent from 26, i.e. 26 + 44 ÷ 52: 26.8462 (0.05% high)
- Tangent from 27, i.e. 27 − 9 ÷ 54: 26.8333 (0% high)
For √720 the tangent at 27 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 720 is just 9 below 729.
Finding √720 with the Babylonian method
Picture a rectangle with an area of 720 and one side x; the other side must be 720 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √720.
Start from the nearest whole number, 27 (27² = 729):
| Step | Guess x | 720 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 27.0000000000 | 26.6666666667 | 26.8333333333 | 3 |
| 2 | 26.8333333333 | 26.8322981366 | 26.8328157350 | 8 |
| 3 | 26.8328157350 | 26.8328157250 | 26.8328157300 | 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 √720 = 26.8328157300 to every decimal shown.
√720 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √720 the pattern is [26; 1, 4, 1, 52] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √720 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 | 8.3 × 10⁻¹ |
| 27/1 | 27.0000000000 | 1.7 × 10⁻¹ |
| 134/5 | 26.8000000000 | 3.3 × 10⁻² |
| 161/6 | 26.8333333333 | 5.2 × 10⁻⁴ |
| 8,506/317 | 26.8328075710 | 8.2 × 10⁻⁶ |
| 8,667/323 | 26.8328173375 | 1.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 720y² = 1. Its smallest solution in positive whole numbers is x = 161, y = 6.
√720 in geometry and everyday measurements
- 720 square feet is 66.9 m². Laid out as a square — a small house footprint or a lot — it is about 26.83 ft (26 ft 10 in) on a side.
- 720 = 12² + 24², so by the Pythagorean theorem √720 is the diagonal of a 12 × 24 rectangle — and the distance between the points (0, 0) and (12, 24) on a grid.
- Since √720 = 12√5, a length of √720 is exactly 12 copies of the length √5 laid end to end.
Square roots near √720 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √717 | √717 | 26.7769 | No |
| √718 | √718 | 26.7955 | No |
| √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 |
- The cube root of 720 is about 8.962809.
- Because 720 = 4 × 180, the root is twice √180: 2 × 13.416408 ≈ 26.832816.
Frequently asked questions
What is the square root of 720?
The square root of 720 is 12√5 in simplest radical form, which is about 26.8328157300. The negative root, −26.832816, also squares to 720.
Is the square root of 720 rational or irrational?
Irrational. 720 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 √720 be simplified?
Yes. The largest perfect square dividing 720 is 144, so √720 = √144 × √5 = 12√5.
What is √720 rounded to two decimal places?
√720 ≈ 26.83 to two decimal places (26.8 to one, 26.833 to three). Check: 26.83² = 719.8489, close to 720.