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.
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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