That Mathematical expression that seems tough !

Added: 2026-05-07

[00:00:01]Hello, welcome back once again.

[00:00:03]Today we have this interesting math problem.

[00:00:07]We're going to find the value for I raised to the power of I.

[00:00:11]Here, I here known as the iota is equal to the square root of -1, which is an imaginary unit.

[00:00:21]I is equal to square root of -1, which implies that I squared is equal to -1.

[00:00:34]Here we're going to find out if an imaginary unit raised to the power of an imaginary unit is going to be real or complex. So, let's get started.

[00:00:45]Now, let us recall E raised to the power of I pi is equal to -1, according to Leonhard Euler.

[00:01:01]Now, let us take the square root on both sides.

[00:01:05]Why we're taking the square root is because we want to make sure that the right hand side will turn out to be I, because square root of -1 is equal to I.

[00:01:14]So, here we get the square root of E raised to the power of I pi is equal to the square root of -1.

[00:01:26]Now, on the left hand side let's apply this property.

[00:01:29]We know the nth root of A to the power of B is equal to A raised to the power of B divided by N.

[00:01:39]The left hand side turns out to be E raised to the power of I pi divided by 2, which is equal to I.

[00:01:52]So, here since this is equal to I, then I raised to the power of I is going to be equal to from the base replace with E raised to the power of I pi divided by 2, then raise it to the power of I.

[00:02:12]Now, let's apply this property.

[00:02:14]We know that A raised to the power of B in bracket raised to the power of C is equal to A to the power of B multiplied by C.

[00:02:23]This will become E raised to the power of I pi divided by 2 multiplied by I, which is equal to E raised to the power of pi by 2 multiplied by I squared.

[00:02:42]Now, we know I squared is equal to -1, so this is equal to E raised to the power of pi by 2 multiplied by -1, which is equal to E raised to the power of -pi by 2.

[00:03:00]So, therefore we say that I raised to the power of I is real, which is equal to E raised to the power of -pi by 2, which is a real number.

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