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

Thread starter exclamaforte
Start date Aug 29, 2010
Tags

Root

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?

Mathematics news on Phys.org

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!

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.