√53 at a glance
- Exact value
- √53
- Decimal (10 places)
- 7.2801098893
- Rounded
- 7.3 · 7.28 · 7.280
- Perfect square?
- No — between 7² and 8²
- Rational?
- Irrational
- Both square roots
- ±7.280110
- Prime factorization
- 53
- Cube root
- 3.756286
How to simplify √53
53 is a prime number, so its only factors are 1 and 53. There is no perfect-square factor to pull out, which means √53 is already in its simplest radical form.
The square root of any prime is irrational. If √53 were a fraction a/b in lowest terms, then a² = 53b², so 53 would divide a — and then 53 would divide b too, contradicting “lowest terms.” That is why the decimal 7.2801098893 is only a rounded value.
Where √53 sits between perfect squares
49 = 7² and 64 = 8² are the nearest perfect squares, so √53 lies between 7 and 8. 53 is 4 above 49 and 11 below 64, so the root is closer to 7.
- Straight line between 49 and 64: 7.2667 (0.18% low)
- Tangent from 7, i.e. 7 + 4 ÷ 14: 7.2857 (0.08% high)
- Tangent from 8, i.e. 8 − 11 ÷ 16: 7.3125 (0.44% high)
For √53 the tangent at 7 wins, missing by only 0.0056. Tangent estimates shine when the number sits close to a perfect square — here 53 is just 4 above 49.
Finding √53 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 53: following the tangent line down to zero simplifies to averaging x with 53 ÷ x.
Start from the nearest whole number, 7 (7² = 49):
| Step | Guess x | 53 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 7.0000000000 | 7.5714285714 | 7.2857142857 | 2 |
| 2 | 7.2857142857 | 7.2745098039 | 7.2801120448 | 5 |
| 3 | 7.2801120448 | 7.2801077337 | 7.2801098893 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √53 = 7.2801098893 to every decimal shown.
√53 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √53 the pattern is [7; 3, 1, 1, 3, 14] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √53 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 |
|---|---|---|
| 7/1 | 7.0000000000 | 2.8 × 10⁻¹ |
| 22/3 | 7.3333333333 | 5.3 × 10⁻² |
| 29/4 | 7.2500000000 | 3.0 × 10⁻² |
| 51/7 | 7.2857142857 | 5.6 × 10⁻³ |
| 182/25 | 7.2800000000 | 1.1 × 10⁻⁴ |
| 2,599/357 | 7.2801120448 | 2.2 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 53y² = 1. Its smallest solution in positive whole numbers is x = 66,249, y = 9,100. Because the period is odd, the equation with −1 on the right also has a solution: 182² − 53 × 25² = −1.
√53 in geometry and everyday measurements
- A square room or garden bed covering 53 square feet measures about 7.28 ft (7 ft 3 in) along each wall.
- 53 = 2² + 7², so by the Pythagorean theorem √53 is the diagonal of a 2 × 7 rectangle — and the distance between the points (0, 0) and (2, 7) on a grid.
Square roots near √53 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √50 | 5√2 | 7.0711 | No |
| √51 | √51 | 7.1414 | No |
| √52 | 2√13 | 7.2111 | No |
| √53 | √53 | 7.2801 | No |
| √54 | 3√6 | 7.3485 | No |
| √55 | √55 | 7.4162 | No |
| √56 | 2√14 | 7.4833 | No |
- The cube root of 53 is about 3.756286.
- Four times the radicand doubles the root: √212 = 2 × √53 ≈ 14.56022.
Frequently asked questions
What is the square root of 53?
The square root of 53 is √53, about 7.2801098893. The negative root, −7.280110, also squares to 53.
Is the square root of 53 rational or irrational?
Irrational. 53 is not a perfect square — it falls between 49 and 64 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √53 be simplified?
No. 53 is prime, so there is no perfect square to take out of the radical.
What is √53 rounded to two decimal places?
√53 ≈ 7.28 to two decimal places (7.3 to one, 7.280 to three). Check: 7.28² = 52.9984, close to 53.