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

Algebra ->  Rational-functions -> SOLUTION: {{{ matrix(2,1,"", 2(x-1)^(3/4)+4=36) }}}       Log On


   

Question 707147: +matrix%282%2C1%2C%22%22%2C%0D%0A%0D%0A%0D%0A2%28x-1%29%5E%283%2F4%29%2B4=36%29+

Found 2 solutions by jsmallt9, Edwin McCravy:
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
+2%28x-1%29%5E%283%2F4%29%2B4=36+
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:
+2%28x-1%29%5E%283%2F4%29=32+
Next we'll eliminate the 2 by dividing both sides by 2:
+%28x-1%29%5E%283%2F4%29=16+
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):
+%28%28x-1%29%5E%283%2F4%29%29%5E%284%2F3%29=16%5E%284%2F3%29+
On the left we get the exponent of 1 we were looking for:
+x-1+=+16%5E%284%2F3%29+
The only thing left to eliminate on the left side is the 1. Adding 1 to each side:
+x+=+1%2B16%5E%284%2F3%29+
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 x%5E%28m%2Fn%29+=+root%28n%2C+x%5Em%29+=+%28root%28n%2C+x%29%29%5Em. I'm going to use the first form:
+x+=+1%2Broot%283%2C+16%5E4%29+
This radical will simplify if there are any perfect cube factors in 16%5E4. Since 8 is a perfect cube and since 8 is a factor of 16 we definitely have perfect cube factors in 16%5E4:
+x+=+1%2Broot%283%2C+%288%2A2%29%5E4%29+
+x+=+1%2Broot%283%2C+%288%2A2%29%2A%288%2A2%29%2A%288%2A2%29%2A%288%2A2%29%29+
+x+=+1%2Broot%283%2C+8%2A8%2A8%2A8%2A%282%2A2%2A2%29%2A2%29+
+x+=+1%2Broot%283%2C+8%2A8%2A8%2A8%2A8%2A2%29+
Now we can use a property of radicals, root%28n%2C+a%2Ab%29+=+root%28n%2C+a%29%2Aroot%28n%2C+b%29, to separate each of the factors into its own cube root:

Each of the cube roots of the perfect cubes will simplify:
+x+=+1%2B2%2A2%2A2%2A2%2A2%2Aroot%283%2C+2%29+
which simplifies to:
[Correction: 2*2*2*2*2 is 32 (as the other tutor showed) not 16 as in my initial solution.]
+x+=+1%2B32%2Aroot%283%2C+2%29+
This is an exact expression for the solution in simplified radical form.

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
The other tutor went into a lot of detail after she
got here.  

+x+=+1%2Broot%283%2C16%5E4%29+

Maybe you'd like this better:

Write the 16 as 2%5E4

+x+=+1%2Broot%283%2C%282%5E4%29%5E4%29+

Then multiply the two exponents 4·4 = 16

+x+=+1%2Broot%283%2C2%5E16%29+

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

+x+=+1%2Broot%283%2C2%5E%2815%2B1%29%29+

Then write 2%5E%2815%2B1%29 as 2%5E15%2A2%5E1 or just 2%5E15%2A2

+x+=+1%2Broot%283%2C2%5E15%2A2%29%29+

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

Then take the 2%5E15 out of the cube root as a 2%5E5 in front
of the radical:

+x+=+1%2B2%5E5%2Aroot%283%2C2%29+

Then write the 2%5E5 as 32

x=1%2B32%2Aroot%283%2C2%29

Edwin