 2022-07-27

I have taken the time to understand exponents and make sesnseof them but I am stuck on rational exponets!
Someone tells me that $4{◻}^{\frac{1}{2}}=\sqrt{4}$ BUT why?
Why is that so and where is the intuition behinde this. Howdid someone come up with this? What is the logic involved instating this.
I can understand the logic behinde $\alpha {◻}^{0}=1$ butthen out of no where $\alpha {◻}^{\frac{1}{n}}=\sqrt[n]{a}$ "becauseit just so happenes that way"(they say) without any realexplanation to why its so. Can someone provide me with the logicbehind this statment.
PS: To learn math is memoization but to understand it isintuition from the mind. yermarvg

Expert

Hi dedesigns,
I think you are asking a very important question. Ipersonally do not care for the form .
While it is useful in some cases, I have always found it easier todeal with rational exponentsIn the form .
Someone really smart is going to come along and give you a greatexplanation, but until then, you will have to bear with my poorattempt.
For now, let's go back a bit to whole number exponents. I am goingto use the number 4 for all examples.
We say that :
We also say that
In addition, we have always held that any whole number can bewritten as a fraction
So, we have to be able to write as . For this to be true, we must get thesame answer. is read as the square root of 4quadrupled. (You could write this as any fraction that is =2 andyou should get the same answer)
Or in the other notation
Rational exponents really are not at all different from wholenumber exponents.
If the rules hold for whole number exponents, they have to hold forfractional exponents also.
So let's go up a few powers using fractional(rational)exponents.
(This is the same as 41).
(This is the same as 42).
We could also do this same exercise using cube roots, but I hopeyou see the idea.
The symbology of this: is not as important as understandingfractional (rational) exponents and how they directly relate towhole number exponents and the relationship between multiplicationand exponents.

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