SOLUTION: Simplify {{{root(3,40a^3b^6c^8)}}}

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Question 148124: Simplify root%283%2C40a%5E3b%5E6c%5E8%29
Found 2 solutions by jim_thompson5910, vleith:
Answer by jim_thompson5910(21685) About Me  (Show Source):
You can put this solution on YOUR website!
root%283%2C40a%5E3b%5E6c%5E8%29 Start with the given expression.


root%283%2C8%2A5a%5E3b%5E6c%5E8%29 Factor 40 into 8%2A5. Notice how 8 is a perfect cube (ie 8=2%5E3).


root%283%2C8%2A5a%5E3b%5E3%2Ab%5E3c%5E8%29 Factor b%5E6 into b%5E3%2Ab%5E3. Once again, notice how b%5E3 is a perfect cube.

root%283%2C8%2A5a%5E3b%5E3%2Ab%5E3c%5E8%29 Factor c%5E8 into c%5E3%2Ac%5E3%2Ac%5E2. Once again, notice how c%5E3 is a perfect cube.


I'm breaking the expression into perfect cubes so that when I take the cube root of these cubes, I'll be left with the expression itself. Since b%5E3 is a perfect cube, this means that root%283%2Cb%5E3%29=b. Also, since c%5E3 is a perfect cube, this means that root%283%2Cc%5E3%29=c.


root%283%2C8%29%2Aroot%283%2C5%29%2Aroot%283%2Ca%5E3%29%2Aroot%283%2Cb%5E3%29%2Aroot%283%2Cb%5E3%29%2Aroot%283%2Cc%5E3%29%2Aroot%283%2Cc%5E3%29%2Aroot%283%2Cc%5E2%29 Break up the root.


2%2Aroot%283%2C5%29%2Aa%2Ab%2Ab%2Ac%2Ac%2Aroot%283%2Cc%5E2%29 Take the cube root of the perfect cubes to get just the base of the expression.

2ab%5E2c%5E2%2Aroot%283%2C5c%5E2%29 Recombine any roots leftover and multiply


So root%283%2C40a%5E3b%5E6c%5E8%29 simplifies to 2ab%5E2c%5E2%2Aroot%283%2C5c%5E2%29


In other words, root%283%2C40a%5E3b%5E6c%5E8%29=2ab%5E2c%5E2%2Aroot%283%2C5c%5E2%29.

Answer by vleith(2517) About Me  (Show Source):