SOLUTION: Simplify {{{root(3,162a^10b^5)/root(3,6a^8b^4)}}}

Algebra ->  Radicals -> SOLUTION: Simplify {{{root(3,162a^10b^5)/root(3,6a^8b^4)}}}      Log On


   

Question 126506: Simplify

root%283%2C162a%5E10b%5E5%29%2Froot%283%2C6a%5E8b%5E4%29
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
root%283%2C162a%5E10b%5E5%29%2Froot%283%2C6a%5E8b%5E4%29 Start with the given expression


First let's simplify the numerator root%283%2C162a%5E10b%5E5%29


Start with the given expression


Break up the roots

3%2Aroot%283%2C6%29%2Aa%5E3%2Aroot%283%2Ca%29%2Ab%2Aroot%283%2Cb%5E2%29 Simplify root%283%2C162%29 to get 3%2Aroot%283%2C6%29. Simplify root%283%2Ca%5E10%29 to get a%5E3%2Aroot%283%2Ca%29. Simplify root%283%2Cb%5E5%29 to get b%2Aroot%283%2Cb%5E2%29



3a%5E3b%2Aroot%283%2C6ab%5E2%29 Multiply and combine the roots


So root%283%2C162a%5E10b%5E5%29 simplifies to 3a%5E3b%2Aroot%283%2C6ab%5E2%29

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Now let's simplify the denominator root%283%2C6a%5E8b%5E4%29


Start with the given expression


Break up the roots

root%283%2C6%29%2Aa%5E2%2Aroot%283%2Ca%5E2%29%2Ab%2Aroot%283%2Cb%29 Simplify root%283%2Ca%5E8%29 to get a%5E2%2Aroot%283%2Ca%5E2%29. Simplify root%283%2Cb%5E4%29 to get b%2Aroot%283%2Cb%29



a%5E2b%2Aroot%283%2C6a%5E2b%29 Multiply and combine the roots

So root%283%2C6a%5E8b%5E4%29 simplifies to a%5E2b%2Aroot%283%2C6a%5E2b%29





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So now our expression becomes

%283a%5E3b%2Aroot%283%2C6ab%5E2%29%29%2F%28a%5E2b%2Aroot%283%2C6a%5E2b%29%29



%283a%2Aroot%283%2C6ab%5E2%29%29%2F%28root%283%2C6a%5E2b%29%29 Divide %283a%5E3b%29%2F%28a%5E2b%29 to get 3a


3a%2Aroot%283%2C%286ab%5E2%29%2F%286a%5E2b%29%29 Combine the roots


3a%2Aroot%283%2Cb%2Fa%29 Divide %286ab%5E2%29%2F%286a%5E2b%29 to get b%2Fa




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Answer:


So root%283%2C162a%5E10b%5E5%29%2Froot%283%2C6a%5E8b%5E4%29 simplifies to 3a%2Aroot%283%2Cb%2Fa%29



In other words, root%283%2C162a%5E10b%5E5%29%2Froot%283%2C6a%5E8b%5E4%29=3a%2Aroot%283%2Cb%2Fa%29