√417 at a glance
- Exact value
- √417
- Decimal (10 places)
- 20.4205778567
- Rounded
- 20.4 · 20.42 · 20.421
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.420578
- Prime factorization
- 3 × 139
- Cube root
- 7.470999
How to simplify √417
The prime factorization of 417 is 3 × 139. Every prime appears only once, so there is no pair to bring outside the radical — √417 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 417, 3 and 139 appear an odd number of times, so √417 is irrational and 20.4205778567 is a rounded value.
Where √417 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √417 lies between 20 and 21. 417 is 17 above 400 and 24 below 441, so the root is closer to 20.
- Straight line between 400 and 441: 20.4146 (0.03% low)
- Tangent from 20, i.e. 20 + 17 ÷ 40: 20.4250 (0.02% high)
- Tangent from 21, i.e. 21 − 24 ÷ 42: 20.4286 (0.04% high)
For √417 the tangent at 20 wins, missing by only 0.0044. Tangent estimates shine when the number sits close to a perfect square — here 417 is just 17 above 400.
Finding √417 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 417: following the tangent line down to zero simplifies to averaging x with 417 ÷ x.
Start from the nearest whole number, 20 (20² = 400):
| Step | Guess x | 417 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 20.8500000000 | 20.4250000000 | 2 |
| 2 | 20.4250000000 | 20.4161566707 | 20.4205783354 | 6 |
| 3 | 20.4205783354 | 20.4205773780 | 20.4205778567 | 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 √417 = 20.4205778567 to every decimal shown.
√417 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √417 the pattern is [20; 2, 2, 1, 1, 1, 5, 4, 1, 12, 1, 4, 5, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √417 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 |
|---|---|---|
| 20/1 | 20.0000000000 | 4.2 × 10⁻¹ |
| 41/2 | 20.5000000000 | 7.9 × 10⁻² |
| 102/5 | 20.4000000000 | 2.1 × 10⁻² |
| 143/7 | 20.4285714286 | 8.0 × 10⁻³ |
| 245/12 | 20.4166666667 | 3.9 × 10⁻³ |
| 388/19 | 20.4210526316 | 4.7 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 417y² = 1. Its smallest solution in positive whole numbers is x = 85,322,647, y = 4,178,268.
√417 in geometry and everyday measurements
- A square garage floor of 417 square feet measures about 20.42 ft (20 ft 5 in) per side, and its corner-to-corner diagonal is √834 ≈ 28.9 ft.
- 417 is not a sum of two whole-number squares — the prime factor 3 and 139 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √417 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 20 box, because 1² + 4² + 20² = 417.
Square roots near √417 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √414 | 3√46 | 20.3470 | No |
| √415 | √415 | 20.3715 | No |
| √416 | 4√26 | 20.3961 | No |
| √417 | √417 | 20.4206 | No |
| √418 | √418 | 20.4450 | No |
| √419 | √419 | 20.4695 | No |
| √420 | 2√105 | 20.4939 | No |
- The cube root of 417 is about 7.470999.
- Squaring undoes the root: (√417)² = 417, while 417² = 173,889 — the number whose square root is 417.
Frequently asked questions
What is the square root of 417?
The square root of 417 is √417, about 20.4205778567. The negative root, −20.420578, also squares to 417.
Is the square root of 417 rational or irrational?
Irrational. 417 is not a perfect square — it falls between 400 and 441 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √417 be simplified?
No. 417 = 3 × 139 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √417 rounded to two decimal places?
√417 ≈ 20.42 to two decimal places (20.4 to one, 20.421 to three). Check: 20.42² = 416.9764, close to 417.