Question 54715
{{{sqrt((12x^3)/5)}}} 
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Break everything into prime factors
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{{{sqrt(((2)(2)(3)(x)(x)(x))/5)}}}
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Write (2)·(2) as (2)² and (x)·(x) as (x)²
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{{{sqrt(((2)^2(3)(x)^2(x))/5)}}}
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Take the (2)² out in front of the radical as just 2
Take the (x)² out in front of the radical as just |x|
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{{{2abs(x)sqrt(((3)(x))/5)}}}
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Rationalize the denominator by multiplying under the
radical by 5/5:
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{{{2abs(x)sqrt((((3)(x))/5)(5/5))}}}
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OR
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{{{2abs(x)sqrt(  ((15)(x))/((5)(5))  )}}}
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Write the (5)·(5) as (5)²
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{{{2abs(x)sqrt(  ((15)(x))/((5)^2)  )}}}
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Take (5)² out in front of the radical as 5, and
since it came from the bottom, put it under the 2|x|.
The absolute value bars are necessary because the original
problem is a square root and even root radicals may never
represent a negative number, so we need absolute
values around the x in case x turned out to be
negative.
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{{{(2abs(x)/5)sqrt(  (15)(x)  )}}}
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This can be written as
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{{{  (2abs(x)sqrt(  15x  ))/5}}}
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Edwin