Square Root of 43
The square root of 43 is approximately 6.557439, or exactly √43 in simplified radical form. No, 43 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √43 sits
6² = 36 and 7² = 49, and 43 falls between them, so √43 is between 6 and 7. It's closer to 7 (7 above 36, 6 below 49), which matches the decimal 6.557439.
| k | k² | Compared to 43 |
|---|---|---|
| 4 | 16 | 27 below |
| 5 | 25 | 18 below |
| 6 | 36 | 7 below |
| 7 | 49 | 6 above |
| 8 | 64 | 21 above |
| 9 | 81 | 38 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 6), divide 43 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 43 ÷ x | New guess (x + 43÷x) ÷ 2 |
|---|---|---|---|
| 1 | 6 | 7.166667 | 6.583333 |
| 2 | 6.583333 | 6.531646 | 6.557489 |
| 3 | 6.557489 | 6.557388 | 6.557439 |
| 4 | 6.557439 | 6.557439 | 6.557439 |
After 4 passes the guess is 6.557439, already matching √43 = 6.557439 to six decimal places.
Exact form from the prime factors
43 = 43. No prime appears twice, so nothing can be pulled out of the root -- √43 is already in simplest form.
Frequently Asked Questions
What is the square root of 43?
The square root of 43 is approximately 6.557439, or exactly √43 in simplified radical form.
Is 43 a perfect square?
No, 43 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √43 between?
Between 6 and 7, because 6² = 36 and 7² = 49.
What are the square roots of the numbers right next to 43?
√42 is about 6.4807, and √44 is about 6.6332. 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 √43 squared?
Exactly 43 -- squaring undoes the square root. Squaring the rounded decimal 6.557439 gives 43, which is why the radical form √43 is the exact answer and the decimal is an approximation.
What is the negative square root of 43?
−6.557439. Every positive number has two square roots, one positive and one negative, because (−6.557439)² is also 43. The radical sign √ by convention means the positive one.
More Number Theory
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