<PRE><B>Question 629523</B>
{{{sqrt(48x^3y^5)/sqrt(18x^5)}}}
While there are manyu ways to to do, the way I like to simplify expressions like this, a square root over a square root, is:&lt;ol&gt;&lt;li&gt;Use the {{{root(a, p)/root(a, q) =root(a, p/q)}}} property of radicals change the fraction of square roots into the square root of a fraction.&lt;/li&gt;&lt;li&gt;Reduce the fraction. (If you&#39;re clever and you know what the next step is, then you may choose not to reduce the fraction fully. I&#39;ll show you what I mean later.)&lt;/li&gt;&lt;li&gt;If there is still a denominator, then...&lt;ol&gt;&lt;li&gt;Make it a perfect square.&lt;/li&gt;&lt;li&gt;Use the {{{roott(a, p)/root(a, q) =root(a, p/q)}}} property again, this time in reverse, to create a fraction of square roots.&lt;/li&gt;&lt;/ol&gt;&lt;li&gt;Simplify any remaining square roots.&lt;/li&gt;&lt;/ol&gt;Let&#39;s see this in action:&lt;br&gt;
1. Merge the square roots
{{{sqrt(48x^3y^5/18x^5)}}}&lt;br&gt;
2. Reduce
{{{sqrt(8y^5/3x^2)}}}&lt;br&gt;
3.1 Make the denominator a perfect square:
{{{sqrt((8y^5/3x^2)(3/3))}}}
{{{sqrt(24y^5/9x^2)}}}&lt;br&gt;
3.2 Split the square root
{{{sqrt(24y^5)/sqrt(9x^2)}}}&lt;br&gt;
4. Simplify
{{{sqrt(4*y^2*y^2*6y)/3x}}}
{{{(sqrt(4)*sqrt(y^2)*sqrt(y^2)*sqrt(6y))/3x}}}
{{{(2*y*y*sqrt(6y))/3x}}}
{{{(2y^2*sqrt(6y))/3x}}}&lt;br&gt;
P.S. Now I&#39;m going to repeat the first two steps using &quot;clever&quot; reducing:
1. Merge the square roots
{{{sqrt(48x^3y^5/18x^5)}}}&lt;br&gt;
2. Reduce
I&#39;m still going to reduce the x&#39;s as before. But instead of dividing the numerator and denominator by 6 like I did above, I am going to just divide by 2:
{{{sqrt(24y^5/9x^2)}}}
The fraction is not fully reduced, But look at the denominator! It is now a perfect square. So by thinking ahead we can skip step 3.1.</PRE>