√70 at a glance
- Exact value
- √70
- Decimal (10 places)
- 8.3666002653
- Rounded
- 8.4 · 8.37 · 8.367
- Perfect square?
- No — between 8² and 9²
- Rational?
- Irrational
- Both square roots
- ±8.366600
- Prime factorization
- 2 × 5 × 7
- Cube root
- 4.121285
How to simplify √70
The prime factorization of 70 is 2 × 5 × 7. Every prime appears only once, so there is no pair to bring outside the radical — √70 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 70, 2, 5 and 7 appear an odd number of times, so √70 is irrational and 8.3666002653 is a rounded value.
Where √70 sits between perfect squares
64 = 8² and 81 = 9² are the nearest perfect squares, so √70 lies between 8 and 9. 70 is 6 above 64 and 11 below 81, so the root is closer to 8.
- Straight line between 64 and 81: 8.3529 (0.16% low)
- Tangent from 8, i.e. 8 + 6 ÷ 16: 8.3750 (0.1% high)
- Tangent from 9, i.e. 9 − 11 ÷ 18: 8.3889 (0.27% high)
For √70 the tangent at 8 wins, missing by only 0.0084. Tangent estimates shine when the number sits close to a perfect square — here 70 is just 6 above 64.
Finding √70 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 8 (8² = 64):
| Step | Guess x | 70 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 8.0000000000 | 8.7500000000 | 8.3750000000 | 2 |
| 2 | 8.3750000000 | 8.3582089552 | 8.3666044776 | 5 |
| 3 | 8.3666044776 | 8.3665960531 | 8.3666002653 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √70 = 8.3666002653 to every decimal shown.
√70 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √70 the pattern is [8; 2, 1, 2, 1, 2, 16] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √70 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 |
|---|---|---|
| 8/1 | 8.0000000000 | 3.7 × 10⁻¹ |
| 17/2 | 8.5000000000 | 1.3 × 10⁻¹ |
| 25/3 | 8.3333333333 | 3.3 × 10⁻² |
| 67/8 | 8.3750000000 | 8.4 × 10⁻³ |
| 92/11 | 8.3636363636 | 3.0 × 10⁻³ |
| 251/30 | 8.3666666667 | 6.6 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 70y² = 1. Its smallest solution in positive whole numbers is x = 251, y = 30.
√70 in geometry and everyday measurements
- A square room or garden bed covering 70 square feet measures about 8.37 ft (8 ft 4 in) along each wall.
- 70 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √70 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 6 box, because 3² + 5² + 6² = 70.
Square roots near √70 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √67 | √67 | 8.1854 | No |
| √68 | 2√17 | 8.2462 | No |
| √69 | √69 | 8.3066 | No |
| √70 | √70 | 8.3666 | No |
| √71 | √71 | 8.4261 | No |
| √72 | 6√2 | 8.4853 | No |
| √73 | √73 | 8.5440 | No |
- The cube root of 70 is about 4.121285.
- Four times the radicand doubles the root: √280 = 2 × √70 ≈ 16.733201.
Frequently asked questions
What is the square root of 70?
The square root of 70 is √70, about 8.3666002653. The negative root, −8.366600, also squares to 70.
Is the square root of 70 rational or irrational?
Irrational. 70 is not a perfect square — it falls between 64 and 81 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √70 be simplified?
No. 70 = 2 × 5 × 7 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √70 rounded to two decimal places?
√70 ≈ 8.37 to two decimal places (8.4 to one, 8.367 to three). Check: 8.37² = 70.0569, close to 70.