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 $sqrt(x^2) = abs(x)$ ,

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

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