Square Root of 26
The square root of 26 is approximately 5.09902, or exactly √26 in simplified radical form. No, 26 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √26 sits
5² = 25 and 6² = 36, and 26 falls between them, so √26 is between 5 and 6. It's closer to 5 (1 above 25, 10 below 36), which matches the decimal 5.09902.
| k | k² | Compared to 26 |
|---|---|---|
| 3 | 9 | 17 below |
| 4 | 16 | 10 below |
| 5 | 25 | 1 below |
| 6 | 36 | 10 above |
| 7 | 49 | 23 above |
| 8 | 64 | 38 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 5), divide 26 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 26 ÷ x | New guess (x + 26÷x) ÷ 2 |
|---|---|---|---|
| 1 | 5 | 5.2 | 5.1 |
| 2 | 5.1 | 5.098039 | 5.09902 |
| 3 | 5.09902 | 5.099019 | 5.09902 |
| 4 | 5.09902 | 5.09902 | 5.09902 |
After 4 passes the guess is 5.09902, already matching √26 = 5.09902 to six decimal places.
Exact form from the prime factors
26 = 2 × 13. No prime appears twice, so nothing can be pulled out of the root -- √26 is already in simplest form.
Frequently Asked Questions
What is the square root of 26?
The square root of 26 is approximately 5.09902, or exactly √26 in simplified radical form.
Is 26 a perfect square?
No, 26 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √26 between?
Between 5 and 6, because 5² = 25 and 6² = 36.
What are the square roots of the numbers right next to 26?
√25 is exactly 5, and √27 is about 5.1962. 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 √26 squared?
Exactly 26 -- squaring undoes the square root. Squaring the rounded decimal 5.09902 gives 26, which is why the radical form √26 is the exact answer and the decimal is an approximation.
What is the negative square root of 26?
−5.09902. Every positive number has two square roots, one positive and one negative, because (−5.09902)² is also 26. The radical sign √ by convention means the positive one.
More Number Theory
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