SOLUTION: {{{ matrix(2,1,"", 2(x-1)^(3/4)+4=36) }}}

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Question 707147:

Found 2 solutions by jsmallt9, Edwin McCravy:
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!

To solve for x wed will "peel away" everything but the x from the left side. First we get rid of the extra term by subtracting 4 from each side:

Next we'll eliminate the 2 by dividing both sides by 2:

Next we eliminate the exponent. This is a little easier than it may seem at first if you realize that you're not actually eliminating the exponent. Every number/expression has an exponent! It's just that if the exponent is a 1 we usually don't bother writing it. So what we are really doing is changing the exponent into a 1. How can we change an exponent from 3/4 to 1? How do we change exponents in any way? We have several rules which tell us how exponents can be changed properly. The fastest, easiest way to change the exponent from 3/4 into a 1 is to use the power of a power rule. We're going to raise both sides of the equation to whatever power which will end up turning that 3/4 into a 1. The power of a power rule says that we should multiply the exponents. So 3/4 times what is 1? If you understand reciprocals you know the answer. Multiplying reciprocals always results in a 1! So we will raise each side of the equation by the reciprocal of 3/4 (which is 4/3):

On the left we get the exponent of 1 we were looking for:

The only thing left to eliminate on the left side is the 1. Adding 1 to each side:

This may be an acceptable answer. But perhaps you (or your teacher) would prefer the answer in simplified radical form (since the fractional exponent on the right indicates a root of some kind). First let's write the expression with a radical. In general . I'm going to use the first form:

This radical will simplify if there are any perfect cube factors in . Since 8 is a perfect cube and since 8 is a factor of 16 we definitely have perfect cube factors in :




Now we can use a property of radicals, , to separate each of the factors into its own cube root:

Each of the cube roots of the perfect cubes will simplify:

which simplifies to:
[Correction: 2*2*2*2*2 is 32 (as the other tutor showed) not 16 as in my initial solution.]

This is an exact expression for the solution in simplified radical form.
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
The other tutor went into a lot of detail after she
got here.  



Maybe you'd like this better:

Write the 16 as 



Then multiply the two exponents 44 = 16



The index of the root is 3,
the exponent of 2 is 16

So write the exponent in terms of its nearest multiple
of the index 3 which does not exceed 16.  So write the 16 
exponent as 15+1



Then write  as  or just 



Then divide the exponent 15 by the index of the radical 3 and get 5

Then take the  out of the cube root as a  in front
of the radical:



Then write the  as 



Edwin
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