√551 at a glance
- Exact value
- √551
- Decimal (10 places)
- 23.4733891886
- Rounded
- 23.5 · 23.47 · 23.473
- Perfect square?
- No — between 23² and 24²
- Rational?
- Irrational
- Both square roots
- ±23.473389
- Prime factorization
- 19 × 29
- Cube root
- 8.198175
How to simplify √551
The prime factorization of 551 is 19 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √551 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 551, 19 and 29 appear an odd number of times, so √551 is irrational and 23.4733891886 is a rounded value.
Where √551 sits between perfect squares
529 = 23² and 576 = 24² are the nearest perfect squares, so √551 lies between 23 and 24. 551 is 22 above 529 and 25 below 576, so the root is closer to 23.
- Straight line between 529 and 576: 23.4681 (0.02% low)
- Tangent from 23, i.e. 23 + 22 ÷ 46: 23.4783 (0.02% high)
- Tangent from 24, i.e. 24 − 25 ÷ 48: 23.4792 (0.02% high)
For √551 the tangent at 23 wins, missing by only 0.0049. Tangent estimates shine when the number sits close to a perfect square — here 551 is just 22 above 529.
Finding √551 with the Babylonian method
If a guess is too big, 551 ÷ 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√551) in one step.
Start from the nearest whole number, 23 (23² = 529):
| Step | Guess x | 551 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 23.0000000000 | 23.9565217391 | 23.4782608696 | 2 |
| 2 | 23.4782608696 | 23.4685185185 | 23.4733896940 | 6 |
| 3 | 23.4733896940 | 23.4733886832 | 23.4733891886 | 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 √551 = 23.4733891886 to every decimal shown.
√551 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √551 the pattern is [23; 2, 8, 1, 8, 2, 46] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √551 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 |
|---|---|---|
| 23/1 | 23.0000000000 | 4.7 × 10⁻¹ |
| 47/2 | 23.5000000000 | 2.7 × 10⁻² |
| 399/17 | 23.4705882353 | 2.8 × 10⁻³ |
| 446/19 | 23.4736842105 | 3.0 × 10⁻⁴ |
| 3,967/169 | 23.4733727811 | 1.6 × 10⁻⁵ |
| 8,380/357 | 23.4733893557 | 1.7 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 551y² = 1. Its smallest solution in positive whole numbers is x = 8,380, y = 357.
√551 in geometry and everyday measurements
- A square garage floor of 551 square feet measures about 23.47 ft (23 ft 6 in) per side, and its corner-to-corner diagonal is √1102 ≈ 33.2 ft.
- 551 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √551 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √551 as its space diagonal.
Square roots near √551 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √548 | 2√137 | 23.4094 | No |
| √549 | 3√61 | 23.4307 | No |
| √550 | 5√22 | 23.4521 | No |
| √551 | √551 | 23.4734 | No |
| √552 | 2√138 | 23.4947 | No |
| √553 | √553 | 23.5160 | No |
| √554 | √554 | 23.5372 | No |
- The cube root of 551 is about 8.198175.
- Squaring undoes the root: (√551)² = 551, while 551² = 303,601 — the number whose square root is 551.
Frequently asked questions
What is the square root of 551?
The square root of 551 is √551, about 23.4733891886. The negative root, −23.473389, also squares to 551.
Is the square root of 551 rational or irrational?
Irrational. 551 is not a perfect square — it falls between 529 and 576 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √551 be simplified?
No. 551 = 19 × 29 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √551 rounded to two decimal places?
√551 ≈ 23.47 to two decimal places (23.5 to one, 23.473 to three). Check: 23.47² = 550.8409, close to 551.