Square Root of 50
The square root of 50 is approximately 7.071068, or exactly 5√2 in simplified radical form. No, 50 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √50 sits
7² = 49 and 8² = 64, and 50 falls between them, so √50 is between 7 and 8. It's closer to 7 (1 above 49, 14 below 64), which matches the decimal 7.071068.
| k | k² | Compared to 50 |
|---|---|---|
| 5 | 25 | 25 below |
| 6 | 36 | 14 below |
| 7 | 49 | 1 below |
| 8 | 64 | 14 above |
| 9 | 81 | 31 above |
| 10 | 100 | 50 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 50 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 50 ÷ x | New guess (x + 50÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 7.142857 | 7.071429 |
| 2 | 7.071429 | 7.070707 | 7.071068 |
| 3 | 7.071068 | 7.071068 | 7.071068 |
| 4 | 7.071068 | 7.071068 | 7.071068 |
After 4 passes the guess is 7.071068, already matching √50 = 7.071068 to six decimal places.
Exact form from the prime factors
50 = 2 × 5^2. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √50 = 5√2. That's the exact value; the decimal 7.071068 is a rounding of it.
Frequently Asked Questions
What is the square root of 50?
The square root of 50 is approximately 7.071068, or exactly 5√2 in simplified radical form.
Is 50 a perfect square?
No, 50 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √50 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 50?
√49 is exactly 7, and √51 is about 7.1414. 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 √50 squared?
Exactly 50 -- squaring undoes the square root. Squaring the rounded decimal 7.071068 gives 50, which is why the radical form 5√2 is the exact answer and the decimal is an approximation.
What is the negative square root of 50?
−7.071068. Every positive number has two square roots, one positive and one negative, because (−7.071068)² is also 50. The radical sign √ by convention means the positive one.
More Number Theory
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