Square Root of 48
The square root of 48 is approximately 6.928203, or exactly 4√3 in simplified radical form. No, 48 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √48 sits
6² = 36 and 7² = 49, and 48 falls between them, so √48 is between 6 and 7. It's closer to 7 (12 above 36, 1 below 49), which matches the decimal 6.928203.
| k | k² | Compared to 48 |
|---|---|---|
| 4 | 16 | 32 below |
| 5 | 25 | 23 below |
| 6 | 36 | 12 below |
| 7 | 49 | 1 above |
| 8 | 64 | 16 above |
| 9 | 81 | 33 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 6), divide 48 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 48 ÷ x | New guess (x + 48÷x) ÷ 2 |
|---|---|---|---|
| 1 | 6 | 8 | 7 |
| 2 | 7 | 6.857143 | 6.928571 |
| 3 | 6.928571 | 6.927835 | 6.928203 |
| 4 | 6.928203 | 6.928203 | 6.928203 |
After 4 passes the guess is 6.928203, already matching √48 = 6.928203 to six decimal places.
Exact form from the prime factors
48 = 2^4 × 3. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √48 = 4√3. That's the exact value; the decimal 6.928203 is a rounding of it.
Frequently Asked Questions
What is the square root of 48?
The square root of 48 is approximately 6.928203, or exactly 4√3 in simplified radical form.
Is 48 a perfect square?
No, 48 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √48 between?
Between 6 and 7, because 6² = 36 and 7² = 49.
What are the square roots of the numbers right next to 48?
√47 is about 6.8557, and √49 is exactly 7. 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 √48 squared?
Exactly 48 -- squaring undoes the square root. Squaring the rounded decimal 6.928203 gives 48, which is why the radical form 4√3 is the exact answer and the decimal is an approximation.
What is the negative square root of 48?
−6.928203. Every positive number has two square roots, one positive and one negative, because (−6.928203)² is also 48. The radical sign √ by convention means the positive one.
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