√247 at a glance
- Exact value
- √247
- Decimal (10 places)
- 15.7162336455
- Rounded
- 15.7 · 15.72 · 15.716
- Perfect square?
- No — between 15² and 16²
- Rational?
- Irrational
- Both square roots
- ±15.716234
- Prime factorization
- 13 × 19
- Cube root
- 6.274305
How to simplify √247
The prime factorization of 247 is 13 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √247 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 247, 13 and 19 appear an odd number of times, so √247 is irrational and 15.7162336455 is a rounded value.
Where √247 sits between perfect squares
225 = 15² and 256 = 16² are the nearest perfect squares, so √247 lies between 15 and 16. 247 is 22 above 225 and 9 below 256, so the root is closer to 16.
- Straight line between 225 and 256: 15.7097 (0.04% low)
- Tangent from 15, i.e. 15 + 22 ÷ 30: 15.7333 (0.11% high)
- Tangent from 16, i.e. 16 − 9 ÷ 32: 15.7188 (0.02% high)
For √247 the tangent at 16 wins, missing by only 0.0025. Tangent estimates shine when the number sits close to a perfect square — here 247 is just 9 below 256.
Finding √247 with the Babylonian method
If a guess is too big, 247 ÷ 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√247) in one step.
Start from the nearest whole number, 16 (16² = 256):
| Step | Guess x | 247 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 16.0000000000 | 15.4375000000 | 15.7187500000 | 2 |
| 2 | 15.7187500000 | 15.7137176938 | 15.7162338469 | 6 |
| 3 | 15.7162338469 | 15.7162334441 | 15.7162336455 | 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 √247 = 15.7162336455 to every decimal shown.
√247 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √247 the pattern is [15; 1, 2, 1, 1, 9, 1, 9, 1, 1, 2, 1, 30] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √247 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 |
|---|---|---|
| 15/1 | 15.0000000000 | 7.2 × 10⁻¹ |
| 16/1 | 16.0000000000 | 2.8 × 10⁻¹ |
| 47/3 | 15.6666666667 | 5.0 × 10⁻² |
| 63/4 | 15.7500000000 | 3.4 × 10⁻² |
| 110/7 | 15.7142857143 | 1.9 × 10⁻³ |
| 1,053/67 | 15.7164179104 | 1.8 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 247y² = 1. Its smallest solution in positive whole numbers is x = 85,292, y = 5,427.
√247 in geometry and everyday measurements
- A square patio or deck of 247 square feet is about 15.72 ft (15 ft 9 in) on each side, so edging all the way around takes 4 × √247 ≈ 62.9 ft.
- 247 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 √247 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 √247 as its space diagonal.
Square roots near √247 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √244 | 2√61 | 15.6205 | No |
| √245 | 7√5 | 15.6525 | No |
| √246 | √246 | 15.6844 | No |
| √247 | √247 | 15.7162 | No |
| √248 | 2√62 | 15.7480 | No |
| √249 | √249 | 15.7797 | No |
| √250 | 5√10 | 15.8114 | No |
- The cube root of 247 is about 6.274305.
- Four times the radicand doubles the root: √988 = 2 × √247 ≈ 31.432467.
Frequently asked questions
What is the square root of 247?
The square root of 247 is √247, about 15.7162336455. The negative root, −15.716234, also squares to 247.
Is the square root of 247 rational or irrational?
Irrational. 247 is not a perfect square — it falls between 225 and 256 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √247 be simplified?
No. 247 = 13 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √247 rounded to two decimal places?
√247 ≈ 15.72 to two decimal places (15.7 to one, 15.716 to three). Check: 15.72² = 247.1184, close to 247.