Square Root of 24
The square root of 24 is approximately 4.898979, or exactly 2√6 in simplified radical form. No, 24 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √24 sits
4² = 16 and 5² = 25, and 24 falls between them, so √24 is between 4 and 5. It's closer to 5 (8 above 16, 1 below 25), which matches the decimal 4.898979.
| k | k² | Compared to 24 |
|---|---|---|
| 2 | 4 | 20 below |
| 3 | 9 | 15 below |
| 4 | 16 | 8 below |
| 5 | 25 | 1 above |
| 6 | 36 | 12 above |
| 7 | 49 | 25 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 4), divide 24 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 24 ÷ x | New guess (x + 24÷x) ÷ 2 |
|---|---|---|---|
| 1 | 4 | 6 | 5 |
| 2 | 5 | 4.8 | 4.9 |
| 3 | 4.9 | 4.897959 | 4.89898 |
| 4 | 4.89898 | 4.898979 | 4.898979 |
After 4 passes the guess is 4.898979, already matching √24 = 4.898979 to six decimal places.
Exact form from the prime factors
24 = 2^3 × 3. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √24 = 2√6. That's the exact value; the decimal 4.898979 is a rounding of it.
Frequently Asked Questions
What is the square root of 24?
The square root of 24 is approximately 4.898979, or exactly 2√6 in simplified radical form.
Is 24 a perfect square?
No, 24 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √24 between?
Between 4 and 5, because 4² = 16 and 5² = 25.
What are the square roots of the numbers right next to 24?
√23 is about 4.7958, and √25 is exactly 5. 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 √24 squared?
Exactly 24 -- squaring undoes the square root. Squaring the rounded decimal 4.898979 gives 24, which is why the radical form 2√6 is the exact answer and the decimal is an approximation.
What is the negative square root of 24?
−4.898979. Every positive number has two square roots, one positive and one negative, because (−4.898979)² is also 24. The radical sign √ by convention means the positive one.
More Number Theory
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