Is it possible to take the ith root of some number?

Thread starter exclamaforte
Start date Aug 29, 2010
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In summary, any non-negative number can be taken to the ith root, but the result may not always be a real number. The ith root of a number can be calculated by raising the number to the power of 1/i. Negative numbers can also be taken to the ith root, but if i is even, the result will be a complex number. If i is a fraction, the result will be a root within a root. There are some limitations to taking the ith root of a number, including the number being non-negative, i not being 0, and potential for a complex number result if i is even and the number is negative.

Aug 29, 2010

exclamaforte

And if so, could I take the i^th root of i?

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Aug 29, 2010

Petr Mugver

[tex]i^{1/i}=i^{-i}=e^{-i\log i}=e^{-i\log e^{i(\pi/2+2k\pi)}}=e^{-ii(\pi/2+2k\pi)}=e^{(\pi/2+2k\pi)}[/tex]

i has infinitely many ith roots, and they are all real!

Related to Is it possible to take the ith root of some number?

1. Can any number be taken to the ith root?

Yes, any non-negative number can be taken to the ith root. However, the result may not always be a real number depending on the value of i and the number itself.

2. How is the ith root of a number calculated?

The ith root of a number can be calculated by raising the number to the power of 1/i. For example, to find the 5th root of 32, we would calculate 32^(1/5) which equals 2.

3. Can negative numbers be taken to the ith root?

Yes, negative numbers can be taken to the ith root. However, if i is an even number, the result will be a complex number. For example, the 4th root of -16 is 2i (2 times the imaginary unit).

4. What happens if i is a fraction?

If i is a fraction, the result will be a root within a root. For example, the 1/3 root of 27 is equal to the cube root of the cube root of 27, which is 3.

5. Are there any limitations to taking the ith root of a number?

Yes, there are some limitations to taking the ith root of a number. The number must be non-negative, and the value of i cannot be 0. Additionally, if i is an even number and the number is negative, the result will be a complex number.