Square Root of 6
The square root of 6 is approximately 2.44949, or exactly √6 in simplified radical form. No, 6 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √6 sits
2² = 4 and 3² = 9, and 6 falls between them, so √6 is between 2 and 3. It's closer to 2 (2 above 4, 3 below 9), which matches the decimal 2.44949.
| k | k² | Compared to 6 |
|---|---|---|
| 1 | 1 | 5 below |
| 2 | 4 | 2 below |
| 3 | 9 | 3 above |
| 4 | 16 | 10 above |
| 5 | 25 | 19 above |
| 6 | 36 | 30 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 2), divide 6 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 6 ÷ x | New guess (x + 6÷x) ÷ 2 |
|---|---|---|---|
| 1 | 2 | 3 | 2.5 |
| 2 | 2.5 | 2.4 | 2.45 |
| 3 | 2.45 | 2.44898 | 2.44949 |
| 4 | 2.44949 | 2.44949 | 2.44949 |
After 4 passes the guess is 2.44949, already matching √6 = 2.44949 to six decimal places.
Exact form from the prime factors
6 = 2 × 3. No prime appears twice, so nothing can be pulled out of the root -- √6 is already in simplest form.
Frequently Asked Questions
What is the square root of 6?
The square root of 6 is approximately 2.44949, or exactly √6 in simplified radical form.
Is 6 a perfect square?
No, 6 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √6 between?
Between 2 and 3, because 2² = 4 and 3² = 9.
What are the square roots of the numbers right next to 6?
√5 is about 2.2361, and √7 is about 2.6458. 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 √6 squared?
Exactly 6 -- squaring undoes the square root. Squaring the rounded decimal 2.44949 gives 6, which is why the radical form √6 is the exact answer and the decimal is an approximation.
What is the negative square root of 6?
−2.44949. Every positive number has two square roots, one positive and one negative, because (−2.44949)² is also 6. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 6
- GCF Calculator
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