Square Root of 42
The square root of 42 is approximately 6.480741, or exactly √42 in simplified radical form. No, 42 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √42 sits
6² = 36 and 7² = 49, and 42 falls between them, so √42 is between 6 and 7. It's closer to 6 (6 above 36, 7 below 49), which matches the decimal 6.480741.
| k | k² | Compared to 42 |
|---|---|---|
| 4 | 16 | 26 below |
| 5 | 25 | 17 below |
| 6 | 36 | 6 below |
| 7 | 49 | 7 above |
| 8 | 64 | 22 above |
| 9 | 81 | 39 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 6), divide 42 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 42 ÷ x | New guess (x + 42÷x) ÷ 2 |
|---|---|---|---|
| 1 | 6 | 7 | 6.5 |
| 2 | 6.5 | 6.461538 | 6.480769 |
| 3 | 6.480769 | 6.480712 | 6.480741 |
| 4 | 6.480741 | 6.480741 | 6.480741 |
After 4 passes the guess is 6.480741, already matching √42 = 6.480741 to six decimal places.
Exact form from the prime factors
42 = 2 × 3 × 7. No prime appears twice, so nothing can be pulled out of the root -- √42 is already in simplest form.
Frequently Asked Questions
What is the square root of 42?
The square root of 42 is approximately 6.480741, or exactly √42 in simplified radical form.
Is 42 a perfect square?
No, 42 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √42 between?
Between 6 and 7, because 6² = 36 and 7² = 49.
What are the square roots of the numbers right next to 42?
√41 is about 6.4031, and √43 is about 6.5574. 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 √42 squared?
Exactly 42 -- squaring undoes the square root. Squaring the rounded decimal 6.480741 gives 42, which is why the radical form √42 is the exact answer and the decimal is an approximation.
What is the negative square root of 42?
−6.480741. Every positive number has two square roots, one positive and one negative, because (−6.480741)² is also 42. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
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