A fractional exponent is an alternate notation for expressing powers and roots together. Fractional Exponents are also known as Radicals or Rational Exponents.
Table of Contents
Whole Number Exponents
First, look at whole number exponents:
The exponent of a number says how many times to use the number in a multiplication.
Remember that when aa is a positive real number, both of these equations are true:
If the exponent is a fraction?
When you have a fractional exponent, the numerator is the power and the denominator is the root. In the variable. So:
- x is a real number
- a and b are positive real numbers
- a is the power
- b is the root
|Exponent of Fraction||Example|
|An exponent of ½ is square root|
|An exponent of ⅓ is cube root|
|An exponent of ¼ is 4th root|
|And so on!||etc|
It worked for ½, it worked with ¼, in fact it works generally:
x1/n = The n-th Root of x
So we can come up with this
A fractional exponent like 1/n means to take the n-th root:
More Complicated Fractions
What about a fractional exponent like 93/2 ?
That is really saying to do a cube (3) and a square root (1/2), in any order.
A fraction (like m/n) can be broken into two parts:
- a whole number part (m) , and
- a fraction (1/n) part
So, because m/n = m × (1/n) we can do this:
And we get this:
A fractional exponent like m/n means:
|Do the m-th power, then take the n-th root||93/2 = 2√(93) = 2√729 = 27|
|Or Take the n-th root and then do the m-th power||93/2 = (2√9)3 = 33 = 27|