√25 at a glance
- Exact value
- 5
- Decimal (10 places)
- 5
- Rounded
- 5
- Perfect square?
- Yes — 5²
- Rational?
- Rational (whole number)
- Both square roots
- ±5
- Prime factorization
- 5²
- Cube root
- 2.924018
How to simplify √25
Write 25 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√25 = 5 = 5
25 has 2 factor pairs (1 × 25, 5 × 5). The pair with two equal numbers, 5 × 5, 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 25 = 1 + 3 + 5 + 7 + … + 9, the first 5 odd numbers.
Where √25 sits between perfect squares
25 is 5². The perfect squares on either side are 16 (4²) and 36 (6²), so the gaps are 9 below and 11 above — consecutive squares always differ by consecutive odd numbers.
Finding √25 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 25: following the tangent line down to zero simplifies to averaging x with 25 ÷ x.
To show the method at work on a perfect square, start one above the answer, at 6:
| Step | Guess x | 25 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 6.0000000000 | 4.1666666667 | 5.0833333333 | 1 |
| 2 | 5.0833333333 | 4.9180327869 | 5.0006830601 | 3 |
| 3 | 5.0006830601 | 4.9993170332 | 5.0000000467 | 7 |
| 4 | 5.0000000467 | 4.9999999533 | 5.0000000000 | all 10 shown |
The count of correct decimals went 1, 3, 7 and all 10 over 4 steps — roughly doubling each time — until the guess matched √25 = 5.0000000000 to every decimal shown.
√25 in geometry and everyday measurements
- A square tile with an area of 25 square inches has sides exactly 5 in long.
- 25 = 3² + 4², so 3, 4 and 5 form a Pythagorean triple: a right triangle with legs 3 and 4 has a hypotenuse of exactly 5.
Square roots near √25 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √22 | √22 | 4.6904 | No |
| √23 | √23 | 4.7958 | No |
| √24 | 2√6 | 4.8990 | No |
| √25 | 5 | 5.0000 | Yes |
| √26 | √26 | 5.0990 | No |
| √27 | 3√3 | 5.1962 | No |
| √28 | 2√7 | 5.2915 | No |
- The cube root of 25 is about 2.924018.
- Taking the square root twice gives the fourth root: √5 ≈ 2.236068.
- Four times the radicand doubles the root: √100 = 2 × √25 ≈ 10.
Frequently asked questions
What is the square root of 25?
The square root of 25 is 5, because 5 × 5 = 25. Strictly, 25 has two square roots, 5 and −5; the √ symbol means the positive one.
Is the square root of 25 rational or irrational?
Rational. 25 is a perfect square, so its square root is the whole number 5.
Can √25 be simplified?
Yes, all the way to a whole number: √25 = 5.
What is √25 rounded to two decimal places?
√25 is exactly 5, or 5.00 written to two decimal places.