Question 314459
If I understand you correctly, your expression should look like this:


{{{(32^(-1/4))^(-4/5)}}}


Because these is a cropping problem with the formula generator when fractional exponents are used, I will make decimal equivalents to allow you to see what's happening better.


{{{1/4}}} is equivalent to .25


{{{4/5}}} is equivalent to .8


Your expression becomes:


{{{(32^(-.25))^(-.8)}}}


To solve this problem,  you will use two of the rules of exponentiation.


The first rule is {{{x^(-n) = 1/x^n}}}


The second rule is {{{(x^m/y^n)^k = x^(m*k)/y^(n*k)}}}


When you have nested parentheses, you always resolve the inner parentheses before resolving the outer parentheses.


We will follow that rule.


Your expression to be simplified is {{{(32^(-.25))^(-.8)}}}


Using the first rule to resolve the inner parentheses, we get:



{{{(1/32^.25)^(-.8)}}}


Using the first rule again to resolve the remaining expression, we get:



{{{(1/(1/32^.25)^(.8))}}}


Using the second rule, our expression becomes:



{{{1/((1^(.8)/32^(.25*.8)))}}}


This simplifies further to:


{{{1/((1^(.8)/32^(.2)))}}}


Since {{{1^.8 = 1}}} and {{{32^(.2) = 2}}}, then our expression becomes {{{1/(1/2)}}} which is equal to {{{2}}}.


Our expression resolves to 2.


You can confirm this answer is correct by plugging the original expression into your calculator and then plugging the final expression into your calculator to see that you get the same answer.


Use the decimal equivalents of the fractional exponents since they are easier to work with, although you should know how to handle the fractions as well.


Your original expression with the decimal equivalents of the fractional exponents is:


{{{(32^(-.25))^(-.8)}}} which resolves to 2.


Your final expression with the decimal equivalents of the fractional exponents is:


{{{1/((1^(.8)/32^(.2)))}}} which resolves to 2.


Both expressions resolve to 2 which indicates that the simplification process worked as expected.


If I were to do this by hand, the progression would look like this:


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