√407 at a glance
- Exact value
- √407
- Decimal (10 places)
- 20.1742410018
- Rounded
- 20.2 · 20.17 · 20.174
- Perfect square?
- No — between 20² and 21²
- Rational?
- Irrational
- Both square roots
- ±20.174241
- Prime factorization
- 11 × 37
- Cube root
- 7.410795
How to simplify √407
The prime factorization of 407 is 11 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √407 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 407, 11 and 37 appear an odd number of times, so √407 is irrational and 20.1742410018 is a rounded value.
Where √407 sits between perfect squares
400 = 20² and 441 = 21² are the nearest perfect squares, so √407 lies between 20 and 21. 407 is 7 above 400 and 34 below 441, so the root is closer to 20.
- Straight line between 400 and 441: 20.1707 (0.02% low)
- Tangent from 20, i.e. 20 + 7 ÷ 40: 20.1750 (0% high)
- Tangent from 21, i.e. 21 − 34 ÷ 42: 20.1905 (0.08% high)
For √407 the tangent at 20 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 407 is just 7 above 400.
Finding √407 with the Babylonian method
If a guess is too big, 407 ÷ 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√407) in one step.
Start from the nearest whole number, 20 (20² = 400):
| Step | Guess x | 407 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 20.0000000000 | 20.3500000000 | 20.1750000000 | 3 |
| 2 | 20.1750000000 | 20.1734820322 | 20.1742410161 | 7 |
| 3 | 20.1742410161 | 20.1742409876 | 20.1742410018 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √407 = 20.1742410018 to every decimal shown.
√407 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √407 the pattern is [20; 5, 1, 2, 1, 5, 40] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √407 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 | 1.7 × 10⁻¹ |
| 101/5 | 20.2000000000 | 2.6 × 10⁻² |
| 121/6 | 20.1666666667 | 7.6 × 10⁻³ |
| 343/17 | 20.1764705882 | 2.2 × 10⁻³ |
| 464/23 | 20.1739130435 | 3.3 × 10⁻⁴ |
| 2,663/132 | 20.1742424242 | 1.4 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 407y² = 1. Its smallest solution in positive whole numbers is x = 2,663, y = 132.
√407 in geometry and everyday measurements
- A square garage floor of 407 square feet measures about 20.17 ft (20 ft 2 in) per side, and its corner-to-corner diagonal is √814 ≈ 28.5 ft.
- 407 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √407 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 √407 as its space diagonal.
Square roots near √407 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √404 | 2√101 | 20.0998 | No |
| √405 | 9√5 | 20.1246 | No |
| √406 | √406 | 20.1494 | No |
| √407 | √407 | 20.1742 | No |
| √408 | 2√102 | 20.1990 | No |
| √409 | √409 | 20.2237 | No |
| √410 | √410 | 20.2485 | No |
- The cube root of 407 is about 7.410795.
- Squaring undoes the root: (√407)² = 407, while 407² = 165,649 — the number whose square root is 407.
Frequently asked questions
What is the square root of 407?
The square root of 407 is √407, about 20.1742410018. The negative root, −20.174241, also squares to 407.
Is the square root of 407 rational or irrational?
Irrational. 407 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 √407 be simplified?
No. 407 = 11 × 37 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √407 rounded to two decimal places?
√407 ≈ 20.17 to two decimal places (20.2 to one, 20.174 to three). Check: 20.17² = 406.8289, close to 407.