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e2m2a
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- Is the odd root of an even number always an irrational number?
Is the odd root of an even number always an irrational number? For example the 7th root or the 11th root, etc. of an even number.
Is 2 the only exception?jedishrfu said:What about the even number 128? the 7th root is 2.
Of course not consider every even or odd positive integer to the 7th power will be its own 7th root.e2m2a said:Is 2 the only exception?
No.e2m2a said:Is the odd root of an even number always an irrational number?
And I have proven that this is the only possible case.mathman said:Looks trivial - no. Take any even number to odd power. It will be even and its odd root will be number you started with.
No, the odd root of an even number can sometimes be a rational number. For example, the square root of 4 is 2, which is both an even number and a rational number.
Yes, an example of an even number with an odd root that is irrational is the square root of 2. The square root of 2 is approximately 1.41421356 and it is neither an even number nor a rational number.
The proof for this statement is based on the irrationality of the square root of 2. It can be shown that if the square root of 2 is irrational, then its odd root (cubic root, fifth root, etc.) will also be irrational. This is because if the odd root of an even number is rational, then it can be simplified to a fraction, which would also imply that the square root of 2 is rational, contradicting its irrationality.
No, there is no specific pattern to determine if the odd root of an even number is irrational. It depends on the individual number and its factors. However, as mentioned before, if the square root of 2 is irrational, then its odd root will also be irrational.
Yes, the odd root of an even number can be a complex number. This can happen when the even number has a negative value. For example, the cube root of -8 is -2, which is both an odd root of an even number and a complex number.