Easier Formula for Rational Exponents?

During my Math II class today, I was looking at yesterdays homework for Rational Exponents and how to simplify them, 16 to the 5/2 power, etc. I was looking through it and got curious and began to simplify a problem, 8 to the 5/3 power, 32 of course, however while I was doing it I got curious and started working backwards, through this I reached a simpler version of the problem being 8([3rd root of 8] to the 5-3 power) [Looks simpler on paper]. I then reverted this to the base formula from b to the m/n power to be b([nth root of b] to the m-n power). In this form you can plug in any base combined with the index and the base's m and turn it into a problem which only needs to be simplified. I am curious if anyone has thought of this yet and if so, where does it originate. I did this with various numbers -1, 1, 2, 3 (simple ones) and saw that it worked with all of them. Let me know if anyone has seen this before please!

#Algebra

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$