√64 at a glance
- Exact value
- 8
- Decimal (10 places)
- 8
- Rounded
- 8
- Perfect square?
- Yes — 8²
- Rational?
- Rational (whole number)
- Both square roots
- ±8
- Prime factorization
- 2⁶
- Cube root
- 4
How to simplify √64
Write 64 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√64 = 2³ = 8
64 has 4 factor pairs (1 × 64, 2 × 32, 4 × 16, 8 × 8). The pair with two equal numbers, 8 × 8, is what makes it a perfect square — and why it has an odd number of factors.
Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 64 = 1 + 3 + 5 + 7 + … + 15, the first 8 odd numbers.
Where √64 sits between perfect squares
64 is 8². The perfect squares on either side are 49 (7²) and 81 (9²), so the gaps are 15 below and 17 above — consecutive squares always differ by consecutive odd numbers.
Finding √64 with the Babylonian method
Picture a rectangle with an area of 64 and one side x; the other side must be 64 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √64.
To show the method at work on a perfect square, start one above the answer, at 9:
| Step | Guess x | 64 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 9.0000000000 | 7.1111111111 | 8.0555555556 | 1 |
| 2 | 8.0555555556 | 7.9448275862 | 8.0001915709 | 3 |
| 3 | 8.0001915709 | 7.9998084337 | 8.0000000023 | 8 |
| 4 | 8.0000000023 | 7.9999999977 | 8.0000000000 | all 10 shown |
The count of correct decimals went 1, 3, 8 and all 10 over 4 steps — roughly doubling each time — until the guess matched √64 = 8.0000000000 to every decimal shown.
√64 in geometry and everyday measurements
- A square room or garden bed covering 64 square feet measures exactly 8 ft (8 ft) along each wall.
- 64 is not a sum of two whole-number squares — no pair of squares adds up to it — so √64 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 √64 as its space diagonal.
Square roots near √64 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √61 | √61 | 7.8102 | No |
| √62 | √62 | 7.8740 | No |
| √63 | 3√7 | 7.9373 | No |
| √64 | 8 | 8.0000 | Yes |
| √65 | √65 | 8.0623 | No |
| √66 | √66 | 8.1240 | No |
| √67 | √67 | 8.1854 | No |
- The cube root of 64 is exactly 4 — 64 is a perfect cube as well (4³), which makes it a perfect sixth power (2⁶).
- Taking the square root twice gives the fourth root: √8 ≈ 2.828427.
- Four times the radicand doubles the root: √256 = 2 × √64 ≈ 16.
Frequently asked questions
What is the square root of 64?
The square root of 64 is 8, because 8 × 8 = 64. Strictly, 64 has two square roots, 8 and −8; the √ symbol means the positive one.
Is the square root of 64 rational or irrational?
Rational. 64 is a perfect square, so its square root is the whole number 8.
Can √64 be simplified?
Yes, all the way to a whole number: √64 = 8.
What is √64 rounded to two decimal places?
√64 is exactly 8, or 8.00 written to two decimal places.