Square Root of 5
The square root of 5 is approximately 2.236068, or exactly √5 in simplified radical form. No, 5 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √5 sits
2² = 4 and 3² = 9, and 5 falls between them, so √5 is between 2 and 3. It's closer to 2 (1 above 4, 4 below 9), which matches the decimal 2.236068.
| k | k² | Compared to 5 |
|---|---|---|
| 1 | 1 | 4 below |
| 2 | 4 | 1 below |
| 3 | 9 | 4 above |
| 4 | 16 | 11 above |
| 5 | 25 | 20 above |
| 6 | 36 | 31 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 2), divide 5 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 5 ÷ x | New guess (x + 5÷x) ÷ 2 |
|---|---|---|---|
| 1 | 2 | 2.5 | 2.25 |
| 2 | 2.25 | 2.222222 | 2.236111 |
| 3 | 2.236111 | 2.236025 | 2.236068 |
| 4 | 2.236068 | 2.236068 | 2.236068 |
After 4 passes the guess is 2.236068, already matching √5 = 2.236068 to six decimal places.
Exact form from the prime factors
5 = 5. No prime appears twice, so nothing can be pulled out of the root -- √5 is already in simplest form.
Frequently Asked Questions
What is the square root of 5?
The square root of 5 is approximately 2.236068, or exactly √5 in simplified radical form.
Is 5 a perfect square?
No, 5 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √5 between?
Between 2 and 3, because 2² = 4 and 3² = 9.
What are the square roots of the numbers right next to 5?
√4 is exactly 2, and √6 is about 2.4495. 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 √5 squared?
Exactly 5 -- squaring undoes the square root. Squaring the rounded decimal 2.236068 gives 5, which is why the radical form √5 is the exact answer and the decimal is an approximation.
What is the negative square root of 5?
−2.236068. Every positive number has two square roots, one positive and one negative, because (−2.236068)² is also 5. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 5
- GCF Calculator
- LCM Calculator