Square Root of 46
The square root of 46 is approximately 6.78233, or exactly √46 in simplified radical form. No, 46 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √46 sits
6² = 36 and 7² = 49, and 46 falls between them, so √46 is between 6 and 7. It's closer to 7 (10 above 36, 3 below 49), which matches the decimal 6.78233.
| k | k² | Compared to 46 |
|---|---|---|
| 4 | 16 | 30 below |
| 5 | 25 | 21 below |
| 6 | 36 | 10 below |
| 7 | 49 | 3 above |
| 8 | 64 | 18 above |
| 9 | 81 | 35 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 6), divide 46 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 46 ÷ x | New guess (x + 46÷x) ÷ 2 |
|---|---|---|---|
| 1 | 6 | 7.666667 | 6.833333 |
| 2 | 6.833333 | 6.731707 | 6.78252 |
| 3 | 6.78252 | 6.78214 | 6.78233 |
| 4 | 6.78233 | 6.78233 | 6.78233 |
After 4 passes the guess is 6.78233, already matching √46 = 6.78233 to six decimal places.
Exact form from the prime factors
46 = 2 × 23. No prime appears twice, so nothing can be pulled out of the root -- √46 is already in simplest form.
Frequently Asked Questions
What is the square root of 46?
The square root of 46 is approximately 6.78233, or exactly √46 in simplified radical form.
Is 46 a perfect square?
No, 46 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √46 between?
Between 6 and 7, because 6² = 36 and 7² = 49.
What are the square roots of the numbers right next to 46?
√45 is about 6.7082, and √47 is about 6.8557. 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 √46 squared?
Exactly 46 -- squaring undoes the square root. Squaring the rounded decimal 6.78233 gives 46, which is why the radical form √46 is the exact answer and the decimal is an approximation.
What is the negative square root of 46?
−6.78233. Every positive number has two square roots, one positive and one negative, because (−6.78233)² is also 46. The radical sign √ by convention means the positive one.
More Number Theory
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