Square Root of 32
The square root of 32 is approximately 5.656854, or exactly 4√2 in simplified radical form. No, 32 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √32 sits
5² = 25 and 6² = 36, and 32 falls between them, so √32 is between 5 and 6. It's closer to 6 (7 above 25, 4 below 36), which matches the decimal 5.656854.
| k | k² | Compared to 32 |
|---|---|---|
| 3 | 9 | 23 below |
| 4 | 16 | 16 below |
| 5 | 25 | 7 below |
| 6 | 36 | 4 above |
| 7 | 49 | 17 above |
| 8 | 64 | 32 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 5), divide 32 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 32 ÷ x | New guess (x + 32÷x) ÷ 2 |
|---|---|---|---|
| 1 | 5 | 6.4 | 5.7 |
| 2 | 5.7 | 5.614035 | 5.657018 |
| 3 | 5.657018 | 5.656691 | 5.656854 |
| 4 | 5.656854 | 5.656854 | 5.656854 |
After 4 passes the guess is 5.656854, already matching √32 = 5.656854 to six decimal places.
Exact form from the prime factors
32 = 2^5. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √32 = 4√2. That's the exact value; the decimal 5.656854 is a rounding of it.
Frequently Asked Questions
What is the square root of 32?
The square root of 32 is approximately 5.656854, or exactly 4√2 in simplified radical form.
Is 32 a perfect square?
No, 32 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √32 between?
Between 5 and 6, because 5² = 25 and 6² = 36.
What are the square roots of the numbers right next to 32?
√31 is about 5.5678, and √33 is about 5.7446. 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 √32 squared?
Exactly 32 -- squaring undoes the square root. Squaring the rounded decimal 5.656854 gives 32, which is why the radical form 4√2 is the exact answer and the decimal is an approximation.
What is the negative square root of 32?
−5.656854. Every positive number has two square roots, one positive and one negative, because (−5.656854)² is also 32. The radical sign √ by convention means the positive one.
More Number Theory
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