Square Root of 29
The square root of 29 is approximately 5.385165, or exactly √29 in simplified radical form. No, 29 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √29 sits
5² = 25 and 6² = 36, and 29 falls between them, so √29 is between 5 and 6. It's closer to 5 (4 above 25, 7 below 36), which matches the decimal 5.385165.
| k | k² | Compared to 29 |
|---|---|---|
| 3 | 9 | 20 below |
| 4 | 16 | 13 below |
| 5 | 25 | 4 below |
| 6 | 36 | 7 above |
| 7 | 49 | 20 above |
| 8 | 64 | 35 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 5), divide 29 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 29 ÷ x | New guess (x + 29÷x) ÷ 2 |
|---|---|---|---|
| 1 | 5 | 5.8 | 5.4 |
| 2 | 5.4 | 5.37037 | 5.385185 |
| 3 | 5.385185 | 5.385144 | 5.385165 |
| 4 | 5.385165 | 5.385165 | 5.385165 |
After 4 passes the guess is 5.385165, already matching √29 = 5.385165 to six decimal places.
Exact form from the prime factors
29 = 29. No prime appears twice, so nothing can be pulled out of the root -- √29 is already in simplest form.
Frequently Asked Questions
What is the square root of 29?
The square root of 29 is approximately 5.385165, or exactly √29 in simplified radical form.
Is 29 a perfect square?
No, 29 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √29 between?
Between 5 and 6, because 5² = 25 and 6² = 36.
What are the square roots of the numbers right next to 29?
√28 is about 5.2915, and √30 is about 5.4772. 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 √29 squared?
Exactly 29 -- squaring undoes the square root. Squaring the rounded decimal 5.385165 gives 29, which is why the radical form √29 is the exact answer and the decimal is an approximation.
What is the negative square root of 29?
−5.385165. Every positive number has two square roots, one positive and one negative, because (−5.385165)² is also 29. The radical sign √ by convention means the positive one.
More Number Theory
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