√4 at a glance
- Exact value
- 2
- Decimal (10 places)
- 2
- Rounded
- 2
- Perfect square?
- Yes — 2²
- Rational?
- Rational (whole number)
- Both square roots
- ±2
- Prime factorization
- 2²
- Cube root
- 1.587401
How to simplify √4
Write 4 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:
√4 = 2 = 2
4 has 2 factor pairs (1 × 4, 2 × 2). The pair with two equal numbers, 2 × 2, 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 4 = 1 + 3, the first 2 odd numbers.
Where √4 sits between perfect squares
4 is 2². The perfect squares on either side are 1 (1²) and 9 (3²), so the gaps are 3 below and 5 above — consecutive squares always differ by consecutive odd numbers.
Finding √4 with the Babylonian method
Picture a rectangle with an area of 4 and one side x; the other side must be 4 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √4.
To show the method at work on a perfect square, start one above the answer, at 3:
| Step | Guess x | 4 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 3.0000000000 | 1.3333333333 | 2.1666666667 | 0 |
| 2 | 2.1666666667 | 1.8461538462 | 2.0064102564 | 2 |
| 3 | 2.0064102564 | 1.9936102236 | 2.0000102400 | 4 |
| 4 | 2.0000102400 | 1.9999897600 | 2.0000000000 | all 10 shown |
The count of correct decimals went 0, 2, 4 and all 10 over 4 steps — roughly doubling each time — until the guess matched √4 = 2.0000000000 to every decimal shown.
√4 in geometry and everyday measurements
- A square tile with an area of 4 square inches has sides exactly 2 in long.
- 4 is not a sum of two whole-number squares — no pair of squares adds up to it — so √4 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 √4 as its space diagonal.
Square roots near √4 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √1 | 1 | 1.0000 | Yes |
| √2 | √2 | 1.4142 | No |
| √3 | √3 | 1.7321 | No |
| √4 | 2 | 2.0000 | Yes |
| √5 | √5 | 2.2361 | No |
| √6 | √6 | 2.4495 | No |
| √7 | √7 | 2.6458 | No |
- The cube root of 4 is about 1.587401.
- Taking the square root twice gives the fourth root: √2 ≈ 1.414214.
- Multiplying the radicand by 100 multiplies the root by 10: √400 = 10 × √4 ≈ 20.
Frequently asked questions
What is the square root of 4?
The square root of 4 is 2, because 2 × 2 = 4. Strictly, 4 has two square roots, 2 and −2; the √ symbol means the positive one.
Is the square root of 4 rational or irrational?
Rational. 4 is a perfect square, so its square root is the whole number 2.
Can √4 be simplified?
Yes, all the way to a whole number: √4 = 2.
What is √4 rounded to two decimal places?
√4 is exactly 2, or 2.00 written to two decimal places.