Square Root of 7
The square root of 7 is approximately 2.645751, or exactly √7 in simplified radical form. No, 7 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √7 sits
2² = 4 and 3² = 9, and 7 falls between them, so √7 is between 2 and 3. It's closer to 3 (3 above 4, 2 below 9), which matches the decimal 2.645751.
| k | k² | Compared to 7 |
|---|---|---|
| 1 | 1 | 6 below |
| 2 | 4 | 3 below |
| 3 | 9 | 2 above |
| 4 | 16 | 9 above |
| 5 | 25 | 18 above |
| 6 | 36 | 29 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 2), divide 7 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 7 ÷ x | New guess (x + 7÷x) ÷ 2 |
|---|---|---|---|
| 1 | 2 | 3.5 | 2.75 |
| 2 | 2.75 | 2.545455 | 2.647727 |
| 3 | 2.647727 | 2.643777 | 2.645752 |
| 4 | 2.645752 | 2.645751 | 2.645751 |
After 4 passes the guess is 2.645751, already matching √7 = 2.645751 to six decimal places.
Exact form from the prime factors
7 = 7. No prime appears twice, so nothing can be pulled out of the root -- √7 is already in simplest form.
Frequently Asked Questions
What is the square root of 7?
The square root of 7 is approximately 2.645751, or exactly √7 in simplified radical form.
Is 7 a perfect square?
No, 7 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √7 between?
Between 2 and 3, because 2² = 4 and 3² = 9.
What are the square roots of the numbers right next to 7?
√6 is about 2.4495, and √8 is about 2.8284. Square roots increase more slowly as the number gets bigger, which is why consecutive whole numbers have roots that get closer together the higher you go.
What is √7 squared?
Exactly 7 -- squaring undoes the square root. Squaring the rounded decimal 2.645751 gives 7, which is why the radical form √7 is the exact answer and the decimal is an approximation.
What is the negative square root of 7?
−2.645751. Every positive number has two square roots, one positive and one negative, because (−2.645751)² is also 7. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 7
- GCF Calculator
- LCM Calculator