SOLUTION: Hii i need help answering this questions ((x^-2/3)^-1*yx^-5/3)/((yx^-5/4)^1/2*x^-1y^4/3)

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Question 406558: Hii i need help answering this questions ((x^-2/3)^-1*yx^-5/3)/((yx^-5/4)^1/2*x^-1y^4/3)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!

We will using all of the following rules/properties of exponents:
  1. Power of a power: %28a%5Ep%29%5Eq+=+a%5E%28%28p%2Aq%29%29
  2. Power of a product: %28a%2Ab%29%5Ep+=+a%5Ep%2Ab%5Ep
  3. Multiplication: a%5Ep%2Aa%5Eq+=+a%5E%28%28p%2Bq%29%29
  4. Division: a%5Ep%2Fa%5Eq+=+a%5E%28%28p-q%29%29
I will refer to these by their numbers.

To simplify this expression we will, of course, use the order of operations (aka PEMDAS). To save time I am going to simplify the numerator and denominator simultaneously.
Exponents first. Using rule #1 on %28x%5E%28-2%2F3%29%29%5E%28-1%29 and rule #2 on %28y%2Ax%5E%28-5%2F4%29%29%5E%281%2F2%29:

The x%5E%28%28-2%2F3%29%2A%28-1%29%29 simplifies

Using Rule #1 on %28x%5E%28-5%2F4%29%29%5E%281%2F2%29:

The x%5E%28%28-5%2F4%29%2A%281%2F2%29%29 simplifies:

Using the Commutative and Associative Properties to rearrange the factors so the the x's and y's are grouped:

Using rule #3 on %28x%5E%282%2F3%29%2Ax%5E%28-5%2F3%29%29, %28x%5E%28-5%2F8%29%2Ax%5E%28-1%29%29 and %28y%5E%281%2F2%29%2Ay%5E%284%2F3%29%29:

Adding the exponents:

%28x%5E%28-1%29%2Ay%29%2F%28x%5E%28-13%2F8%29%2Ay%5E%2811%2F6%29%29
Using rule #4 to divide the x's and the y's:
x%5E%28%28-1%29-%28-13%2F8%29%29%2Ay%5E%281-%2811%2F6%29%29
Subtracting the exponents:
x%5E%28%28-8%2F8%29-%28-13%2F8%29%29%2Ay%5E%28%286%2F6%29-%2811%2F6%29%29
x%5E%28%28-8%2F8%29%2B%28-13%2F8%29%29%2Ay%5E%28%286%2F6%29%2B%28-11%2F6%29%29
x%5E%28-21%2F8%29%2Ay%5E%28-5%2F6%29
This may be an acceptable answer. Your post did not say anything about a preferred form for the answer.

Sometimes problems like these ask you to express your answer without any negative or zero exponents. If this is the case (and since negative exponents means reciprocals) the answer would be:
1%2F%28x%5E%2821%2F8%29%2Ay%5E%285%2F6%29%29

Sometimes you are asked to express fractional exponents in radical form. If this is the case then your answer would be:
1%2F%28root%288%2C+x%5E21%29%2Aroot%286%2C+y%5E5%29%29
Normally, simplified expressions do not have radicals in denominators. If you need to see this done then post another question: "How do I rationalize the denominator of 1%2F%28root%288%2C+x%5E21%29%2Aroot%286%2C+y%5E5%29%29?