be careful when an imaginary number is raised to a fractional power

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The powers of the imaginary unit i are always intriguing. Here we will examine i^(4/4). Is the result 1 or i? We have to be careful when an imaginary number is raised to a fractional power!
0:00
0:06 is i = 1?
4:23 (i^4)^(1/4) vs (i^(1/4))^4
10:14 (i^3)^(1/4) vs (i^(1/4))^3
13:53 summary on z^(m/n)
15:41 check out Brilliant to learn more!
16:35 bonus part
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[@NatoSkato]

Its like saying 1 = sqrt(1) and then picking -1 for the square root and then saying 1 = -1

[@official-obama]

NEVER calculate roots without a trusted adult's supervision

[@vibby1004]

Step 3 creates 3 extraneous solutions, it’s really cool imo

[@Sam_on_YouTube]

That use of quotes is called "scare quotes." It is a play on words from how they are used when speaking, which is as "air quotes." When you use quotes for something that isn't literally being said it indicates that you are using the term, but not attributing the meaning to yourself. So you are saying it is okay, but you are distancing yourself from the word. You are putting it in quotes so it is not attributed to you, but just to what other people might say.

[@justfrankjustdank2538]

he would look so different and the same without the beard

[@minhmouse]

Bruh. Only...... I

[@tylershepard4269]

As an electrical engineering PhD student, this phenomena is what we refer to as “phase unwrapping.” Where the phase is the argument of the complex number. The wrapped phase always lies between -pi and pi. Once the wrapped phase goes above pi, it “wraps” back around to -pi and continues increasing. To unwrap the phase, look for discontinuities in the phase and add 2*pi to the phase thereafter for each discontinuity encountered.

[@trueberryless]

But this is 3i 😂😂😂, best joke ever...

[@RoderickEtheria]

Mistake is ruling out the complex roots when square rooting.

[@myboatforacar]

I'm scared for the 3% who thought the problem was step 1

[@blackpenredpen]

Good morning!

[@yoavboaz1078]

you can do a real number version of it

[@lunstee]

I find this reminding me of derivatives and indefinite integrals. The integral has an arbitrary constant that the derivative is blind to, so the derivative of the integral of a function is the original function. The integral of the derivative of a function on the other hand, can differ from the original function by a constant.

[@stapler942]

If I'm understanding correctly, this might be partly a consequence of the fact that z^x is not truly invertible, and the reciprocal of an exponent is not a true inverse of that exponent. Even in the real numbers, even powers are not invertible.

[@watwat7050]

It's always such a joy to see you solving these problems and explaining them clearly. Cheers mate!

[@swisssr]

In your second example, the order didn't matter for the ^3/4 because you considered all the possible values for i^3 by writing it in the polar form. But for the first example ^4/4, you just reduced i^4 to 1, instead of writing it in polar form (e^(i*pi*(1/2 + 2*n)))^4 , which gives i (as expected) when raised to 1/4. When you do the fraction i^(1/4) first, it works just because you use the polar form. In summary, as long as you write it in the polar form (not only when the power is fraction, but integer too), it works. The polar form also explains the "real number version" as someone mentioned in another comment:

[@joonjoon2368]

I’m just mesmerized by his marker quick switch

[@damianbla4469]

17:15

[@HopeP0H]

Took me a minute before I realized that there were multiple roots.

[@camrouxbg]

This is so clear! Thanks so much! Really refreshed roots of unity for me, and also refreshed some of those basic exponent "rules" that I hadn't fully internalized.