Fractional Exponents
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Key Questions

I will show you what fractional exponents are
Suppose we are asked to simplify this :
#(16)^(1/4)# Basically this means that we have to find the
#4^(th)# root of 16so in the form of a picture it will be like this

The reciprocal of the number associated with the radical is the power needed.
Examples:
#root3(5)=5^(1/3)#
#root7(2)=2^(1/7)# If the radicand (number under the radical sign) has a power in it, the same method still works:
#root4(9^2)=9^(2/4)# This can be simplified to get
#9^(1/2)# . 
#x^(a/b) =rootb(x^a) = (rootb(x))^a# You can just remember this rule, or you can learn about why this is:
fractional exponent
#1/b# So first we're going to look at an expression of the form:
#x^(1/b)# .
To investigate what this means, we need to go from#x to x^(1/b)# and then deduce something from it.#x^1 = x^(b/b) = x^(1/b*b)#
What does multiplication mean? Repeated addition. So we can instead of multiplying by b, adding the number to itself#b# times.
#x^(1/b+1/b+1/b+1/b +...)# (b times)There is a rule you use when multiplying numbers with the same radical: add the exponents. If we reverse this rule, we get:
#x^(1/b)*x^(1/b)*x^(1/b)*x^(1/b)*x^(1/b)...# (b times)Now, we still know that this number is equal to
#x# . So now we have to think a bit. What number, multiplied by itself b times, gives you#x# .
It's the bthroot of#x# =>#x^(1/b)=rootbx# For example:
#8^(1/3)#
If we multiply this by itself 3 times we get:
#8^(1/3)*8^(1/3)*8^(1/3) = 8^(3/3) = 8#
What number multiplied by itself 3 times, gives you 8.
It's of course#root3(8) = 2# What about
#a/b# ?
To know what#x^(a/b)# means, we can further rely on our previous findings:
#x^(a/b) = x^(a*1/b) = x^(1/b+1/b+1/b+1/b...) # (a times)
#= x^(1/b)*x^(1/b)*x^(1/b)...# (a times)Repeated multiplication is equal to exponentiation, so we can write:
#= (x^(1/b))^a = (rootbx)^a# You can also bring the exponent in the root:
#= rootb(x^a)# 
We can rewrite:
#b^{m/n}=root{n}{b^m}#
Example
#3^{5/7}=root{7}{3^5}#
I hope that this was helpful.
Questions
Videos on topic View all (2)
Exponents and Exponential Functions

1Exponential Properties Involving Products

2Exponential Properties Involving Quotients

3Negative Exponents

4Fractional Exponents

5Scientific Notation

6Scientific Notation with a Calculator

7Exponential Growth

8Exponential Decay

9Geometric Sequences and Exponential Functions

10Applications of Exponential Functions