√625 at a glance
- Exact value
- 25
- Decimal (10 places)
- 25
- Rounded
- 25
- Perfect square?
- Yes — 25²
- Rational?
- Rational (whole number)
- Both square roots
- ±25
- Prime factorization
- 5⁴
- Cube root
- 8.549880
How to simplify √625
Write 625 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√625 = 5² = 25
625 has 3 factor pairs (1 × 625, 5 × 125, 25 × 25). The pair with two equal numbers, 25 × 25, 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 625 = 1 + 3 + 5 + 7 + … + 49, the first 25 odd numbers.
Where √625 sits between perfect squares
625 is 25². The perfect squares on either side are 576 (24²) and 676 (26²), so the gaps are 49 below and 51 above — consecutive squares always differ by consecutive odd numbers.
Finding √625 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 625: following the tangent line down to zero simplifies to averaging x with 625 ÷ x.
To show the method at work on a perfect square, start one above the answer, at 26:
| Step | Guess x | 625 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 26.0000000000 | 24.0384615385 | 25.0192307692 | 1 |
| 2 | 25.0192307692 | 24.9807840123 | 25.0000073908 | 5 |
| 3 | 25.0000073908 | 24.9999926092 | 25.0000000000 | all 10 shown |
The count of correct decimals went 1, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √625 = 25.0000000000 to every decimal shown.
√625 in geometry and everyday measurements
- A square garage floor of 625 square feet measures exactly 25 ft (25 ft) per side, and its corner-to-corner diagonal is √1250 ≈ 35.4 ft.
- 625 = 7² + 24², so 7, 24 and 25 form a Pythagorean triple: a right triangle with legs 7 and 24 has a hypotenuse of exactly 25. 625 splits into two squares in 2 ways (7² + 24², 15² + 20²).
Square roots near √625 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √622 | √622 | 24.9399 | No |
| √623 | √623 | 24.9600 | No |
| √624 | 4√39 | 24.9800 | No |
| √625 | 25 | 25.0000 | Yes |
| √626 | √626 | 25.0200 | No |
| √627 | √627 | 25.0400 | No |
| √628 | 2√157 | 25.0599 | No |
- The cube root of 625 is about 8.549880.
- Taking the square root twice gives the fourth root: √25 = 5, so 625 = 5⁴.
- Squaring undoes the root: (√625)² = 625, while 625² = 390,625 — the number whose square root is 625.
Frequently asked questions
What is the square root of 625?
The square root of 625 is 25, because 25 × 25 = 625. Strictly, 625 has two square roots, 25 and −25; the √ symbol means the positive one.
Is the square root of 625 rational or irrational?
Rational. 625 is a perfect square, so its square root is the whole number 25.
Can √625 be simplified?
Yes, all the way to a whole number: √625 = 25.
What is √625 rounded to two decimal places?
√625 is exactly 25, or 25.00 written to two decimal places.