How do you simplify \frac { \sqrt { 18x y ^ { 2} } } { \sqrt { 49x } }?

1 Answer
Nov 3, 2017

(3ysqrt2)/7

Explanation:

Our first step in simplifying this radical expression is to utilize the fact that, for any two numbers a and b:

sqrt(ab)=sqrtasqrtb

With this in mind, we can rewrite the original expression as

sqrt(18xy^2)/sqrt(49x)=(sqrt18sqrtxsqrt(y^2))/(sqrt49sqrtx)

We can then cancel the sqrtx in the numerator and the denominator to obtain

(sqrt18cancel(sqrtx)sqrt(y^2))/(sqrt49cancel(sqrtx))=(sqrt18sqrt(y^2))/sqrt49

The square root of any number squared is simply that number, so we can simplify sqrt(y^2) to y, and since 49 is just 7^2, sqrt49 must be 7. We now have:

(sqrt18y)/7

If we want to simplify sqrt18, we'll need to factor a perfect square out of the 18, and then again take advantage of the property of radicals stated at the beginning of the explanation.

Since 18 is the product of the perfect square 9 and 2, we can rewrite the radical as sqrt(9*2), which is equivalent to sqrt9sqrt2. Since 9 is 3^2, we can simplify sqrt9 to 3; we can't simplify sqrt2 any more, so our simplified radical for sqrt18 would be 3sqrt2.

Putting that back into our expression, we finally get (3sqrt2y)/7, which, depending on your aesthetic taste, you can either leave as-is or rewrite as (3ysqrt2)/7.