SOLUTION: simplify by factoring cube rt of sqrt 48x^8

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Question 211272: simplify by factoring
cube rt of sqrt 48x^8
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, let's simplify sqrt%2848x%5E8%29


sqrt%2848%2Ax%5E8%29 Start with the given expression.


sqrt%2816%2A3%2Ax%5E8%29 Factor 48 into 16%2A3


sqrt%2816%2A3%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2%29 Factor x%5E8 into x%5E2%2Ax%5E2%2Ax%5E2%2Ax%5E2


Break up the square root using the identity sqrt%28A%2AB%29=sqrt%28A%29%2Asqrt%28B%29.


4%2Asqrt%283%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29%2Asqrt%28x%5E2%29 Take the square root of 16 to get 4.


4%2Asqrt%283%29%2Ax%2Ax%2Ax%2Ax Take the square root of x%5E2 to get x.


4x%5E4%2Asqrt%283%29 Rearrange and multiply the terms.


So sqrt%2848%2Ax%5E8%29 simplifies to 4x%5E4%2Asqrt%283%29


In other words, sqrt%2848%2Ax%5E8%29=4x%5E4%2Asqrt%283%29 where every variable is non-negative.


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So root%283%2Csqrt%2848%2Ax%5E8%29%29 simplifies to root%283%2C4x%5E4%2Asqrt%283%29%29


Ie. root%283%2Csqrt%2848%2Ax%5E8%29%29=root%283%2C4x%5E4%2Asqrt%283%29%29 where x%3E=0


Now let's simplify root%283%2C4x%5E4%2Asqrt%283%29%29


root%283%2C4x%5E4%2Asqrt%283%29%29 Start with the given expression.


root%283%2C4x%5E4%29%2Aroot%283%2Csqrt%283%29%29 Break up the root.


root%283%2C4x%2Ax%5E3%29%2Aroot%283%2Csqrt%283%29%29 Factor x%5E4 to get x%2Ax%5E3


root%283%2C4x%29%2Aroot%283%2Cx%5E3%29%2Aroot%283%2Csqrt%283%29%29 Break up the first root.


root%283%2C4x%29%2Ax%2Aroot%283%2Csqrt%283%29%29 Take the cube root of x%5E3 to get x


%284x%29%5E%281%2F3%29%5E%22%22%2Ax%2A%283%5E%281%2F2%29%29%5E%281%2F3%29 Convert to exponential notation.


%284x%29%5E%281%2F3%29%2Ax%2A3%5E%281%2F6%29%5E%22%22 Multiply the exponents.


%284x%29%5E%282%2F6%29%5E%22%22%2Ax%2A3%5E%281%2F6%29 Rewrite the first exponent 1%2F3 as 2%2F6


x%2A%28%284x%29%5E%282%2F6%29%5E%22%22%2A3%5E%281%2F6%29%29 Rearrange the terms.


x%2A%28%284x%29%5E2%2A3%5E1%29%5E%281%2F6%29 Factor out 1%2F6


x%2A%2816x%5E2%2A3%29%5E%281%2F6%29 Square and simplify


x%2A%2848x%5E2%29%5E%281%2F6%29 Multiply


x%2Aroot%286%2C48x%5E2%29 Convert back to radical notation.



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Answer:


So root%283%2Csqrt%2848%2Ax%5E8%29%29 completely simplifies to x%2Aroot%286%2C48x%5E2%29


In other words, root%283%2Csqrt%2848%2Ax%5E8%29%29=x%2Aroot%286%2C48x%5E2%29 where x%3E=0