√453 at a glance
- Exact value
- √453
- Decimal (10 places)
- 21.2837966538
- Rounded
- 21.3 · 21.28 · 21.284
- Perfect square?
- No — between 21² and 22²
- Rational?
- Irrational
- Both square roots
- ±21.283797
- Prime factorization
- 3 × 151
- Cube root
- 7.680086
How to simplify √453
The prime factorization of 453 is 3 × 151. Every prime appears only once, so there is no pair to bring outside the radical — √453 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 453, 3 and 151 appear an odd number of times, so √453 is irrational and 21.2837966538 is a rounded value.
Where √453 sits between perfect squares
441 = 21² and 484 = 22² are the nearest perfect squares, so √453 lies between 21 and 22. 453 is 12 above 441 and 31 below 484, so the root is closer to 21.
- Straight line between 441 and 484: 21.2791 (0.02% low)
- Tangent from 21, i.e. 21 + 12 ÷ 42: 21.2857 (0.01% high)
- Tangent from 22, i.e. 22 − 31 ÷ 44: 21.2955 (0.05% high)
For √453 the tangent at 21 wins, missing by only 0.0019. Tangent estimates shine when the number sits close to a perfect square — here 453 is just 12 above 441.
Finding √453 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 453: following the tangent line down to zero simplifies to averaging x with 453 ÷ x.
Start from the nearest whole number, 21 (21² = 441):
| Step | Guess x | 453 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 21.0000000000 | 21.5714285714 | 21.2857142857 | 2 |
| 2 | 21.2857142857 | 21.2818791946 | 21.2837967402 | 7 |
| 3 | 21.2837967402 | 21.2837965674 | 21.2837966538 | 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 √453 = 21.2837966538 to every decimal shown.
√453 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √453 the pattern is [21; 3, 1, 1, 10, 14, 10, 1, 1, 3, 42] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √453 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 |
|---|---|---|
| 21/1 | 21.0000000000 | 2.8 × 10⁻¹ |
| 64/3 | 21.3333333333 | 5.0 × 10⁻² |
| 85/4 | 21.2500000000 | 3.4 × 10⁻² |
| 149/7 | 21.2857142857 | 1.9 × 10⁻³ |
| 1,575/74 | 21.2837837838 | 1.3 × 10⁻⁵ |
| 22,199/1,043 | 21.2837967402 | 8.6 × 10⁻⁸ |
The same fractions solve Pell’s equation, x² − 453y² = 1. Its smallest solution in positive whole numbers is x = 1,653,751, y = 77,700.
√453 in geometry and everyday measurements
- A square garage floor of 453 square feet measures about 21.28 ft (21 ft 3 in) per side, and its corner-to-corner diagonal is √906 ≈ 30.1 ft.
- 453 is not a sum of two whole-number squares — the prime factor 3 and 151 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √453 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 14 × 16 box, because 1² + 14² + 16² = 453.
Square roots near √453 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √450 | 15√2 | 21.2132 | No |
| √451 | √451 | 21.2368 | No |
| √452 | 2√113 | 21.2603 | No |
| √453 | √453 | 21.2838 | No |
| √454 | √454 | 21.3073 | No |
| √455 | √455 | 21.3307 | No |
| √456 | 2√114 | 21.3542 | No |
- The cube root of 453 is about 7.680086.
- Squaring undoes the root: (√453)² = 453, while 453² = 205,209 — the number whose square root is 453.
Frequently asked questions
What is the square root of 453?
The square root of 453 is √453, about 21.2837966538. The negative root, −21.283797, also squares to 453.
Is the square root of 453 rational or irrational?
Irrational. 453 is not a perfect square — it falls between 441 and 484 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √453 be simplified?
No. 453 = 3 × 151 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √453 rounded to two decimal places?
√453 ≈ 21.28 to two decimal places (21.3 to one, 21.284 to three). Check: 21.28² = 452.8384, close to 453.