SOLUTION: Perform the operations and simplify the following. Express your answers with positive exponents only. {[(3x^-3y^2)^-3] / -6x^3y^-4}^-2

Algebra ->  Exponents -> SOLUTION: Perform the operations and simplify the following. Express your answers with positive exponents only. {[(3x^-3y^2)^-3] / -6x^3y^-4}^-2      Log On


   

Question 386531: Perform the operations and simplify the following. Express your answers with positive exponents only.
{[(3x^-3y^2)^-3] / -6x^3y^-4}^-2
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
%28%28%283x%5E%28-3%29y%5E2%29%5E-3%29%2F%28-6x%5E3y%5E%28-4%29%29%29%5E%28-2%29
With these problems where the answer is supposed to have only positive exponents, I recommend that you make changing negative exponents to positive one the last step. In other words we will go ahead and simplify the expression as much as possible first. Then, if there are any negative exponents left, we change negative exponentes to positive ones.

The first thing to simplify, according to the order of operations (aka PEMDAS) is the numerator of the fraction inside the big parentheses. Using the rule for exponsnts, %28a%2Ab%29%2Aq+=+a%5Eq%2Ab%5Eq, we get:

For the x's and y's in the numerator, the rule is to multiply the exponents:
%28%283%5E%28-3%29%2Ax%5E9%2Ay%5E%28-6%29%29%2F%28-6x%5E3y%5E%28-4%29%29%29%5E%28-2%29
Now we can simplify the fraction. The rule for the exponents when we divide the x's and y's is to subtract the exponents:
%28%283%5E%28-3%29%2Ax%5E%289-3%29%2Ay%5E%28-6-%28-4%29%29%29%29%2F%28-6%29%29%5E%28-2%29
%28%283%5E%28-3%29%2Ax%5E6%2Ay%5E%28-2%29%29%2F%28-6%29%29%5E%28-2%29
Now we can arise this fraction to the -2 power using both %28a%2Fb%29%5Eq+=+a%5Eq%2Fb%5Eq and %28a%2Ab%29%2Aq+=+a%5Eq%2Ab%5Eq:

which simpifies as follows:

%283%5E6%2Ax%5E%28-12%29%2Ay%5E4%29%2F%28-6%29%5E%28-2%29
After all this work, we have fewer negative exponents than when we started. So by not worrying about positive exponents until now we have saved ourselves some work. Using both a%5E%28-n%29+=+1%2Fa%5En and 1%2Fa%5E%28-n%29+=+a%5En we can convert both negative exponents into equivalent expressions with positive exponents:
%283%5E6%2A%281%2Fx%5E12%29%2Ay%5E4%29%2F%281%2F%28-6%29%5E2%29
which simplifies as follows:
%2836%2A3%5E6%2Ay%5E4%29%2Fx%5E12
%2836%2A729%2Ay%5E4%29%2Fx%5E12
%2826244y%5E4%29%2Fx%5E12