SOLUTION: (3x ^2 yz ^3) ^2 (x^4y) ^3

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Question 283873: (3x ^2 yz ^3) ^2 (x^4y) ^3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
To simplify, we're going to use the following identities (equations that are true for all values of the given variable(s))


1. %28ab%29%5Ex=a%5Ex%2Ab%5Ex. Basically, you can either multiply first, then exponentiate, OR, exponentiate and then multiply. (note: this can expand to any number of terms inside the parenthesis. So you could have 50 terms in the parenthesis if you wanted). Example: %283x%29%5E3=%283%29%5E3%2A%28x%29%5E3


2. %28x%5Ey%29%5Ez=x%5E%28y%2Az%29. When you raise an exponential expression to another exponent, you're going to multiply the exponents. Example: %28x%5E2%29%5E3=x%5E%282%2A3%29=x%5E6


3. a%5Ex%2Aa%5Ey=a%5E%28x%2By%29. When you multiply monomials (with a common base), you're going to add the exponents. Example: x%5E3%2Ax%5E5=x%5E%283%2B5%29=x%5E8


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Let's use the identities shown above to simplify. I'll refer to each identity listed by the number associated with them.


%283x%5E2yz%5E3%29%5E2%28x%5E4y%29%5E3 Start with the given expression.


%283%29%5E2%2A%28x%5E2%29%5E2%2A%28y%29%5E2%2A%28z%5E3%29%5E2%28x%5E4%29%5E3%2A%28y%29%5E3 Use identity number 1 (see above) to break up the expression.


Use identity number 2.


%283%29%5E2%2A%28x%5E4%29%2A%28y%29%5E2%2A%28z%5E6%29%2A%28x%5E%2812%29%29%2A%28y%29%5E3 Multiply


%283%29%5E2%2A%28x%5E4%2Ax%5E%2812%29%29%2A%28y%5E2%2Ay%5E3%29%2A%28z%5E6%29 Group the terms with common bases (ie group all the 'x' terms together, etc.)


%283%29%5E2%2A%28x%5E%284%2B12%29%29%2A%28y%5E%282%2B3%29%29%2A%28z%5E6%29 Use identity #3 to multiply the monomials.


%283%29%5E2%2Ax%5E%2816%29%2Ay%5E5%2Az%5E6 Add


9%2Ax%5E16%2Ay%5E5%2Az%5E6 Square 3 to get 3%5E2=3%2A3=9


So %283x%5E2yz%5E3%29%5E2%28x%5E4y%29%5E3 simplifies to 9%2Ax%5E16%2Ay%5E5%2Az%5E6


In other words, %283x%5E2yz%5E3%29%5E2%28x%5E4y%29%5E3=9%2Ax%5E16%2Ay%5E5%2Az%5E6