SOLUTION: How would yiou simplify the following (assuming that the value of each variable is positive): (30x^3)^(1/2)∙√6x^5y^2z^3 I have a test tomorrow and myself and mo

Algebra ->  Radicals -> SOLUTION: How would yiou simplify the following (assuming that the value of each variable is positive): (30x^3)^(1/2)∙√6x^5y^2z^3 I have a test tomorrow and myself and mo      Log On


   

Question 176930: How would yiou simplify the following (assuming that the value of each variable is positive):
(30x^3)^(1/2)∙√6x^5y^2z^3

I have a test tomorrow and myself and most of the people in my algebra class are still extremely confused. Please, please, please, help me with this and explain in full detail if possible! You're help would be very much appreciated!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%2830x%5E3%29%5E%281%2F2%29%2Asqrt%286x%5E5%2Ay%5E2%2Az%5E3%29 Start with the given expression.


sqrt%2830x%5E3%29%2Asqrt%286x%5E5%2Ay%5E2%2Az%5E3%29 Convert to radical notation


sqrt%28%2830x%5E3%29%286x%5E5%2Ay%5E2%2Az%5E3%29%29 Combine the roots using the identity sqrt%28x%29%2Asqrt%28y%29=sqrt%28x%2Ay%29


sqrt%28180%2Ax%5E8%2Ay%5E2%2Az%5E3%29%29 Multiply


sqrt%2836%2A5%2Ax%5E8%2Ay%5E2%2Az%5E3%29%29 Factor 180 into 36%2A5 (note: one factor is a perfect square)


sqrt%2836%2A5%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ay%5E2%2Az%5E3%29%29 Factor x%5E8 into x%5E4%2Ax%5E4=x%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2 (note: the goal is to get "x" with an exponent of 2)


sqrt%2836%2A5%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ay%5E2%2Az%5E2%2Az%29%29 Factor z%5E3 into z%5E3=z%5E2%2Az


Break up the square root.


Take the square root of 36 to get 6


6%2Asqrt%285%29%2Ax%2Ax%2Ax%2Ax%2Asqrt%28y%5E2%29%2Asqrt%28z%5E2%29%2Asqrt%28z%29%29 Take the square root of x%5E2 to get x


6%2Asqrt%285%29%2Ax%2Ax%2Ax%2Ax%2Ay%2Asqrt%28z%5E2%29%2Asqrt%28z%29%29 Take the square root of y%5E2 to get y


6%2Asqrt%285%29%2Ax%2Ax%2Ax%2Ax%2Ay%2Az%2Asqrt%28z%29%29 Take the square root of z%5E2 to get z


6%2Asqrt%285%29%2Ax%5E4%2Ay%2Az%2Asqrt%28z%29%29 Multiply


6%2Ax%5E4%2Ay%2Az%2Asqrt%285z%29%29 Recombine the roots.


===============================================================

Answer:

So the expression

%2830x%5E3%29%5E%281%2F2%29%2Asqrt%286x%5E5%2Ay%5E2%2Az%5E3%29


completely simplifies to


6x%5E4%2Ay%2Az%2Asqrt%285z%29


In other words,

%2830x%5E3%29%5E%281%2F2%29%2Asqrt%286x%5E5%2Ay%5E2%2Az%5E3%29=6x%5E4%2Ay%2Az%2Asqrt%285z%29 where every variable is positive.