Math Arguments

Square Root Exponents

We all know that adding/subtracting exponents corresponds to multiplying/dividing the terms, like this:

\(x^4 * x^7 = x^{11}\)

\(\dfrac{x^{14}}{x^9} = x^5\)

Then negative exponents logically followed: \(x^{-7} = \dfrac{x^2}{x^9}\)

Then \(\dfrac{x^3}{x^3} = x^0 = 1\) logically followed that. 

Additionally, multiplying/dividing the exponents relates to powers/roots

\( {x^4}^2 = x^{4*2} = x^8\)

\( \sqrt{x^6} = x^{6/2} = x^3\)

Thus, a fractional exponent means a radical, depending on the denominator of the exponent.

So here's my question:

What other mathematical aspects of this caught your attention? What do you wonder about the situation?