Square Root of 30
The square root of 30 is approximately 5.477226, or exactly √30 in simplified radical form. No, 30 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √30 sits
5² = 25 and 6² = 36, and 30 falls between them, so √30 is between 5 and 6. It's closer to 5 (5 above 25, 6 below 36), which matches the decimal 5.477226.
| k | k² | Compared to 30 |
|---|---|---|
| 3 | 9 | 21 below |
| 4 | 16 | 14 below |
| 5 | 25 | 5 below |
| 6 | 36 | 6 above |
| 7 | 49 | 19 above |
| 8 | 64 | 34 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 5), divide 30 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 30 ÷ x | New guess (x + 30÷x) ÷ 2 |
|---|---|---|---|
| 1 | 5 | 6 | 5.5 |
| 2 | 5.5 | 5.454545 | 5.477273 |
| 3 | 5.477273 | 5.477178 | 5.477226 |
| 4 | 5.477226 | 5.477226 | 5.477226 |
After 4 passes the guess is 5.477226, already matching √30 = 5.477226 to six decimal places.
Exact form from the prime factors
30 = 2 × 3 × 5. No prime appears twice, so nothing can be pulled out of the root -- √30 is already in simplest form.
Frequently Asked Questions
What is the square root of 30?
The square root of 30 is approximately 5.477226, or exactly √30 in simplified radical form.
Is 30 a perfect square?
No, 30 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √30 between?
Between 5 and 6, because 5² = 25 and 6² = 36.
What are the square roots of the numbers right next to 30?
√29 is about 5.3852, and √31 is about 5.5678. 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 √30 squared?
Exactly 30 -- squaring undoes the square root. Squaring the rounded decimal 5.477226 gives 30, which is why the radical form √30 is the exact answer and the decimal is an approximation.
What is the negative square root of 30?
−5.477226. Every positive number has two square roots, one positive and one negative, because (−5.477226)² is also 30. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
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