Simplifying Fractional Exponents

I’ve never had an intuitive understanding of why you can simplify fractional exponents the same way you would simplify any other fraction. For example, why does 8^(4/6)= 8^(2/3)?

Let’s start with the simplified fractional exponent, 8^(2/3). Another way to look at this exponent is 8^(2/3) = (8²)^1/3.

When working with exponents it always helps me understand what’s actually happening by writing out the individual terms. Here’s another way to look at 8^(2/3):

x = 8^(2/3)
=> rewrite as equivalent statement
x = (8²)^(1/3)
=> raise each side to the 3rd power
x³ = 8²
=> write out exponents
x * x * x = 8 * 8

Using this same approach to notation, if we translate this fractional exponent to an equivalent fraction like 8^(4/6), it looks like the following:

x = 8^(4/6) 
=> follow same steps as above compressed into one step
x * x * x * x * x * x = 8 * 8 * 8 * 8

This does look like an unsimplified version of the first equation we saw for 8^(2/3). By grouping some of the terms together we can algebraically simplify this expression and get to our understanding of why these expressions are equivalent:

x * x * x * x * x * x = 8 * 8 * 8 * 8
=> group terms
(x * x * x) * (x * x * x) = (8 * 8) * (8 * 8)
=> recognize and simplify
(x * x * x)² = (8 * 8)²
=> take square root of both sides
x * x * x = 8 * 8

Once I noticed it was essentially the problem of squaring both sides of the equation it struck me that that makes sense since a way that you could look at 8^(4/6) is raising 8^(2/3) to the 2/2 power:

8^(4/6) = (8 ^(2/3))^(2/2)

2/2 is 1, in the same way that squaring and then taking the square root of a number is the same as doing nothing at all to it!

Math is full of inverse operations, and playing around with these numbers helped me gain a deeper understanding of that relationship between exponents and roots. This helped solidify an intuitive understanding that keeping exponents and roots (the numerator and denominator of a fractional exponent) in the same proportion leads to equivalent statements (i.e. 8^(4/6) = 8^(2/3)).

I wouldn’t be surprised at all if there are errors here, so please let me know if you see something!