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

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

### 4 Answers by Expert Tutors

Parviz F. | Mathematics professor at Community CollegesMathematics professor at Community Colle...
4.8 4.8 (4 lesson ratings) (4)
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Radical an fractional exponents are equivalent:

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

n√ Xn  = X

Then if We have ( Xn1/n

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

X n(1/n)  = X

Then    power of 1/n equals n√ .

Kay G. | ~20 Years Accounting Tutoring Experience~20 Years Accounting Tutoring Experience
4.9 4.9 (32 lesson ratings) (32)
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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.)

ax 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√(ay ) means x root of (ay)

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.

Excellent explanation! However, be careful when working with real number variables:
√(x^2) == |x| while (√(x))^2 = x.
That might depend on who you ask.  If you're implying that the x in the first example could actually be a negative number (which theoretically it could), unlike way back when I was in high school, what I'm seeing taught today is that they don't consider negative as a choice for an answer.  I can find you examples exactly like that, where that exponent is indeed inside the radical and not outside of it.  The use of parenthesis was simply so it was understood that the whole thing was under the radical sign, since we can't draw the top line of it, although I'm not really sure if you have an issue with the parenthesis, or just the fact that the exponent is under the radical sign.
Ebenezer O. | Aerospace Engr & Air Traffic Control Grad For General Ed. TutoringAerospace Engr & Air Traffic Control Gra...
4.6 4.6 (13 lesson ratings) (13)
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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;

x½ is the same as √x

x¼ is the same as 4√x

x¾ is the same as 4√(x) 3

xa/b is the same as b√(x) a

Hope this helps. Cheers!
Jason S. | Science/math--college level and belowScience/math--college level and below
5.0 5.0 (9 lesson ratings) (9)
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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 nth root of any number (x) can also be expressed as: x(1/n) .
Hopefully that helps.