SOLUTION: how do you solve for x^-2/3=1/9

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: how do you solve for x^-2/3=1/9      Log On


   

Question 677194: how do you solve for x^-2/3=1/9
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E%28%28-2%2F3%29%29+=+1%2F9
By the time we're done, our equation will look like:
x = something
I point this out because keeping mind where you're headed can help you figure out what to do. In this case, since the exponent on "x" is 1, all we need to do is find a way to change the exponent on x from -2/3 to 1.

We have several rules which tell us proper ways exponents can be changed. The one we will use is the one that does not require an additional term with "x", the power of a power rule:
%28a%5Ep%29%5Eq+=+a%5E%28p%2Aq%29
We have x%5E%28%28-2%2F3%29%29. By raising both sides to some power the exponent on x will change from -2/3 to -2/3 times whatever power we choose. We just have to choose a power so that -2/3 times that power will be a one. And if you know about reciprocals you may know that the product of reciprocals is always a 1! So we will raise both sides of the equation to the reciprocal of -2/3 power:
%28x%5E%28%28-2%2F3%29%29%29%5E%28%28-3%2F2%29%29+=+%281%2F9%29%5E%28%28-3%2F2%29%29
The left side simplifies as we planned:
x%5E%28%28-2%2F3%29%2A%28-3%2F2%29%29+=+%281%2F9%29%5E%28%28-3%2F2%29%29
x+=+%281%2F9%29%5E%28%28-3%2F2%29%29

All we have left to do is simplify the right side. If you have trouble with negative and/or fractional exponents then I find a special way of factoring the exponents can be helpful:
  1. If the exponent is negative then factor out -1.
  2. If the exponent is a fraction and the numerator is not 1, then factor out the numerator.
Let's see this in action. The exponent on the right is negative so we factor out -1 (in the exponent):
x+=+%281%2F9%29%5E%28%28-1%2A%283%2F2%29%29%29
The exponent has a fraction whose numerator is not a 1 so we factor out the numerator:
x+=+%281%2F9%29%5E%28%28-1%2A3%2A%281%2F2%29%29%29
The reason factoring the exponent helps is that each factor tells us an operation to perform:
  • The -1 tells us that we will have to find a reciprocal.
  • The 3 tells us that we will cube something.
  • The 1/2 tells us that we will find a square root
And the order in which these operations are done doesn't matter! So we can do these things in whatever order we choose!

So what looks easiest to start with? Reciprocal, cube? square root? Cubing 1/9 doesn't like much fun. A square root of 1/9 might be easy. But a reciprocal of 1/9 is not only easy but it will turn 1/9 into a whole number. So that is where I choose to start. As you do an operation, remove the factor of the exponent that told you to do it:
x+=+9%5E%28%283%2A%281%2F2%29%29%29
Cubing a 9 doesn't look as easy as finding a square root of 9. So we'll do the square root next:
x+=+3%5E3
And finally we cube:
x+=+27