Question 74023
Given: {{{(81)*(x^12)*(y^8)*(z^16)}}}
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To find the term that, when raised to the fourth power, results in the above expression,
you can just take the fourth root of the above expression.  One way of writing the fourth root
is to just raise the above expression to the {{{1/4}}} power.  Then you can use the rules
of exponents. When you raise the above to the {{{1/4}}} power, the expression becomes:
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{{{((81)*(x^12)*(y^8)*(z^16))^(1/4)}}}
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next you apply the {{{1/4}}} power to each of the factors in the given expression to get:
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{{{(81)^(1/4)*(x^12)^(1/4)*(y^8)^(1/4)*(z^16)^(1/4)}}}
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The first factor of 81 has a fourth root of 3 because 3*3*3*3 = 81.  So let's replace {{{81^(1/4)}}}
by 3.  When you do the expression is simplified to:
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{{{3*(x^12)^(1/4)*(y^8)^(1/4)*(z^16)^(1/4)}}}
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Then each of the other terms is raised to the {{{1/4}}} power by multiplying its exponent by {{{1/4}}}.
So:
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{{{(x^12)^(1/4) = x^(12/4) = x^3}}} and 
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{{{(y^8)^(1/4) = y^(8/4) = y^2}}}  and
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{{{(z^16)^(1/4) = z^(16/4) = z^4}}}
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Multiplying all these answers together results in:
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{{{3*x^3*y^2*z^4}}}
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And that's the answer. If you reverse the process and raise the answer to the fourth
power, you get back to the original expression.
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Hope this helps you to see how the rules of exponents can be applied to a problem such
as this.  You should be particularly aware that an exponent of {{{1/2}}} is equivalent
to taking the square root, and exponent of {{{1/3}}} is equivalent to taking the cube root,
and, as in this case, an exponent of {{{1/4}}} is equivalent to finding the fourth root ...
and so on.