What is a fractional exponent? How are fractional exponents and radicals related? Do you prefer using fractional exponents or radicals when performing operation

Question

This is the question on my homework. The problem is I am having a lot of issues with math language, Can you please explain?

Answer 1

 Radical an fractional exponents are equivalent:

 

  Consider Roots, that are opposite of the exponents, i.e.

 

  ⁿ√ Xⁿ  = X 

 

        Then if We have ( Xⁿ ) ^1/n

 

          using laws of exponents of raising an exponent to another exponent.

 

           X ^n(1/n)  = X 

 

        Then    power of 1/n equals ⁿ√ .

    

Answer 2

You said you were having issues with the math language.  You've been shown what is equivalent to what, but not an explanation of which is called by which name (the language.)

 
a^x is where x is the exponent
 
So 2³ has an exponent of 3.
 
A fractional exponent is the same thing, except the x is going to be a fraction like:
 
2^(1/3) 
 
Which is 2 to the 1/3rd.  The (1/3) is the exponent and it's a fraction; hence, fractional exponent.
 
A radical is with the √ sign.  So square root, cubed root, etc are radicals.  Like:
 
^x√a means the x root of a.
 
²√16 means the square root of 16 (which is 4).  (Usually just written as √16 with the 2 being understood.  That only works when it's just a 2.)
 
A fractional exponent can be written as a radical.  As already shown:
 
^x√(a^y ) means x root of (a^y)
 
2^(1/3) (fractional exponent) can be written as ³√2 (radical)
 
4^(2/3) (fractional exponent) can be written as ³√(4²) (radical)
 
Note where the numerator and denominator of the fraction in the exponent are when you do it as a radical.  (numerator inside and denominator outside)  When the numerator is 1, you don't write it.  In my example 2^(1/3) = ³√2 note I never put the 1 exponent on the radical.  2¹ = 2 so the 1 is not needed and we leave it off.

 
You have to decide which is easier to solve. :-)  If you're doing it manually, you just about have to turn it into a radical.  (2 multiplied times itself 1/3 times?  Sounds a little weird, huh?)  If you're using a calculator either can be just as easy depending on the calculator.  But I suspect since you're just learning these, you're doing them manually.
 

That's probably a lot very quickly, but once you're getting into radicals, they "assume" you already know how to do the exponents.

Answer 3

Well fractional exponent can be written as radicals and radical can also be written as fractional exponents. It is always easier to solve with radicals than with fractional exponents but in some situations, the latter can be preferred.

 

Here's some examples;

 

x^½ is the same as √x

 

x^¼ is the same as ⁴√x

 

x^¾ is the same as ⁴√(x) ³

 

 

x^a/b is the same as ^b√(x) ^a

 

 

Hope this helps. Cheers!

Answer 4

A fractional exponent is used to represent radical operations in a different way. For example, if you are taking the square root of the number 4, an equivalent expression is 4^(1/2) . If you plug that into a calculator you should get 2 either way. Another example is the cube root of 8 can also be expressed at 8^(1/3) . There is a recognizable pattern:

 

the n^th root of any number (x) can also be expressed as: x^(1/n) .

Hopefully that helps.