√913 at a glance
- Exact value
- √913
- Decimal (10 places)
- 30.2158898595
- Rounded
- 30.2 · 30.22 · 30.216
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.215890
- Prime factorization
- 11 × 83
- Cube root
- 9.701158
How to simplify √913
The prime factorization of 913 is 11 × 83. Every prime appears only once, so there is no pair to bring outside the radical — √913 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 913, 11 and 83 appear an odd number of times, so √913 is irrational and 30.2158898595 is a rounded value.
Where √913 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √913 lies between 30 and 31. 913 is 13 above 900 and 48 below 961, so the root is closer to 30.
- Straight line between 900 and 961: 30.2131 (0.01% low)
- Tangent from 30, i.e. 30 + 13 ÷ 60: 30.2167 (0% high)
- Tangent from 31, i.e. 31 − 48 ÷ 62: 30.2258 (0.03% high)
For √913 the tangent at 30 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 913 is just 13 above 900.
Finding √913 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 913: following the tangent line down to zero simplifies to averaging x with 913 ÷ x.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 913 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 30.4333333333 | 30.2166666667 | 3 |
| 2 | 30.2166666667 | 30.2151130723 | 30.2158898695 | 8 |
| 3 | 30.2158898695 | 30.2158898495 | 30.2158898595 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √913 = 30.2158898595 to every decimal shown.
√913 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √913 the pattern is [30; 4, 1, 1, 1, 2, 1, 1, 6, 7, 2, 2, 19, …] with the block of 32 terms after the semicolon repeating forever (only the first 12 of the 32 are shown). A pattern that never ends is one more proof that √913 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 |
|---|---|---|
| 30/1 | 30.0000000000 | 2.2 × 10⁻¹ |
| 121/4 | 30.2500000000 | 3.4 × 10⁻² |
| 151/5 | 30.2000000000 | 1.6 × 10⁻² |
| 272/9 | 30.2222222222 | 6.3 × 10⁻³ |
| 423/14 | 30.2142857143 | 1.6 × 10⁻³ |
| 1,118/37 | 30.2162162162 | 3.3 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 913y² = 1. Its smallest solution in positive whole numbers is x = 515,734,243,080,407, y = 17,068,312,251,564 — 15 digits for x, even though 913 is small, which is what makes Pell’s equation famous.
√913 in geometry and everyday measurements
- 913 square feet is 84.8 m². Laid out as a square — a small house footprint or a lot — it is about 30.22 ft (30 ft 3 in) on a side.
- 913 is not a sum of two whole-number squares — the prime factor 11 and 83 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √913 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 30 box, because 2² + 3² + 30² = 913.
Square roots near √913 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √910 | √910 | 30.1662 | No |
| √911 | √911 | 30.1828 | No |
| √912 | 4√57 | 30.1993 | No |
| √913 | √913 | 30.2159 | No |
| √914 | √914 | 30.2324 | No |
| √915 | √915 | 30.2490 | No |
| √916 | 2√229 | 30.2655 | No |
- The cube root of 913 is about 9.701158.
- Squaring undoes the root: (√913)² = 913, while 913² = 833,569 — the number whose square root is 913.
Frequently asked questions
What is the square root of 913?
The square root of 913 is √913, about 30.2158898595. The negative root, −30.215890, also squares to 913.
Is the square root of 913 rational or irrational?
Irrational. 913 is not a perfect square — it falls between 900 and 961 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √913 be simplified?
No. 913 = 11 × 83 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √913 rounded to two decimal places?
√913 ≈ 30.22 to two decimal places (30.2 to one, 30.216 to three). Check: 30.22² = 913.2484, close to 913.