√7 at a glance
- Exact value
- √7
- Decimal (10 places)
- 2.6457513111
- Rounded
- 2.6 · 2.65 · 2.646
- Perfect square?
- No — between 2² and 3²
- Rational?
- Irrational
- Both square roots
- ±2.645751
- Prime factorization
- 7
- Cube root
- 1.912931
How to simplify √7
7 is a prime number, so its only factors are 1 and 7. There is no perfect-square factor to pull out, which means √7 is already in its simplest radical form.
The square root of any prime is irrational. If √7 were a fraction a/b in lowest terms, then a² = 7b², so 7 would divide a — and then 7 would divide b too, contradicting “lowest terms.” That is why the decimal 2.6457513111 is only a rounded value.
Where √7 sits between perfect squares
4 = 2² and 9 = 3² are the nearest perfect squares, so √7 lies between 2 and 3. 7 is 3 above 4 and 2 below 9, so the root is closer to 3.
- Straight line between 4 and 9: 2.6000 (1.73% low)
- Tangent from 2, i.e. 2 + 3 ÷ 4: 2.7500 (3.94% high)
- Tangent from 3, i.e. 3 − 2 ÷ 6: 2.6667 (0.79% high)
For √7 the tangent at 3 wins, missing by only 0.0209. Tangent estimates shine when the number sits close to a perfect square — here 7 is just 2 below 9.
Finding √7 with the Babylonian method
If a guess is too big, 7 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√7) in one step.
Start from the nearest whole number, 3 (3² = 9):
| Step | Guess x | 7 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 3.0000000000 | 2.3333333333 | 2.6666666667 | 1 |
| 2 | 2.6666666667 | 2.6250000000 | 2.6458333333 | 4 |
| 3 | 2.6458333333 | 2.6456692913 | 2.6457513123 | 8 |
| 4 | 2.6457513123 | 2.6457513098 | 2.6457513111 | all 10 shown |
The count of correct decimals went 1, 4, 8 and all 10 over 4 steps — roughly doubling each time — until the guess matched √7 = 2.6457513111 to every decimal shown.
√7 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √7 the pattern is [2; 1, 1, 1, 4] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √7 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 |
|---|---|---|
| 2/1 | 2.0000000000 | 6.5 × 10⁻¹ |
| 3/1 | 3.0000000000 | 3.5 × 10⁻¹ |
| 5/2 | 2.5000000000 | 1.5 × 10⁻¹ |
| 8/3 | 2.6666666667 | 2.1 × 10⁻² |
| 37/14 | 2.6428571429 | 2.9 × 10⁻³ |
| 45/17 | 2.6470588235 | 1.3 × 10⁻³ |
The same fractions solve Pell’s equation, x² − 7y² = 1. Its smallest solution in positive whole numbers is x = 8, y = 3.
√7 in geometry and everyday measurements
- A square tile with an area of 7 square inches has sides about 2.646 in long.
- 7 is not a sum of two whole-number squares — 7 is itself a prime that is one less than a multiple of 4, which rules that out — so √7 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 √7 as its space diagonal.
Square roots near √7 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √4 | 2 | 2.0000 | Yes |
| √5 | √5 | 2.2361 | No |
| √6 | √6 | 2.4495 | No |
| √7 | √7 | 2.6458 | No |
| √8 | 2√2 | 2.8284 | No |
| √9 | 3 | 3.0000 | Yes |
| √10 | √10 | 3.1623 | No |
- The cube root of 7 is about 1.912931.
- Multiplying the radicand by 100 multiplies the root by 10: √700 = 10 × √7 ≈ 26.457513.
Frequently asked questions
What is the square root of 7?
The square root of 7 is √7, about 2.6457513111. The negative root, −2.645751, also squares to 7.
Is the square root of 7 rational or irrational?
Irrational. 7 is not a perfect square — it falls between 4 and 9 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √7 be simplified?
No. 7 is prime, so there is no perfect square to take out of the radical.
What is √7 rounded to two decimal places?
√7 ≈ 2.65 to two decimal places (2.6 to one, 2.646 to three). Check: 2.65² = 7.0225, close to 7.