How do you rationalize the denominator and simplify #sqrt((20y)/(5x))#?

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Answer 1 · 2016-03-31T03:00:58

#sqrt({20y}/{5x}) = frac{2 sqrt(xy)}{abs(x)}#

We begin by noting that 20 is divisible by 5. Hence,

#sqrt({20y}/{5x}) = sqrt({4y}/x)#

From the law of indices, we know that

#sqrt({ab}/c) = sqrta sqrt(b/c)# if #a >= 0#

So,

#sqrt({4y}/x) = sqrt4 sqrt(y/x)#

#= 2 sqrt(y/x)#

To rationalize the denominator, we multiply #x# on both the numerator and the denominator.

#2 sqrt(y/x)= 2 sqrt((y xx x)/(x xx x))#

#= 2 sqrt((xy)/(x^2))#

And from the fact #sqrt(x^2) = abs(x)#,

#2 sqrt((xy)/(x^2)) = frac{2 sqrt(xy)}{abs(x)}#