Question 277473
{{{sqrt(a/b)*sqrt(a^2/b^2)}}}
When simplifying expressions of square roots there are two issues to address:<ul><li>Simplifying the square roots by extracting perfect square factors, if any</li><li>Making sure any denominators are rational (i.e. no square roots in denominators.</li></ul>
To accomplish these tasks we often need one or both of the following properties of radicals:<ul><li>{{{root(a, p) * root(a, q) = root(a, p*q)}}}</li><li>{{{root(a, p) / root(a, q) = root(a, p/q)}}}</li></ul>
These properties can and often are used in both "directions". For example the first property lets us change
{{{sqrt(ab^2)}}} into {{{sqrt(a) * sqrt(b^2)}}}
It also lets us change
{{{sqrt(8)*sqrt(2)}}} into {{{sqrt(8*2)}}}<br>
As you probably know already, there are often many ways to solve a problem. This is especially true with problems like this. So there are many ways to reach the simplified expression for this problem. Here's one way:
Use the second property to split the square roots of the fractions into fractions of square roots:
{{{(sqrt(a)/sqrt(b))*(sqrt(a^2)/sqrt(b^2))}}}
Now we'll multiply the two fractions using the first property to multiply the square roots:
{{{sqrt(a^3)/sqrt(b^3)}}}
Next I'll address the issue of rational denominators. We have a square root in the denominator which we cannot have in our final expression. To rationalize a one-term denominator like this we will multiply the numerator and the denominator by whatever square root that will make the denominator turn into a perfect square. The most obvious choice is to multiply by {{{sqrt(b^3)}}}. And this will work. But the best choice is the one that creates the "smallest" perfect square. (It's kind of like adding fractions. You can add 1/2 and 1/4 if you turn both denominators into 8's. But it is better to use denominators of 4.) The best choice is to multiply by {{{sqrt(b)}}}:
{{{(sqrt(a^3)/sqrt(b^3))(sqrt(b)/sqrt(b))}}}
which results in:
{{{sqrt(a^3b)/sqrt(b^4)}}}
The denominator simplifies to:
{{{sqrt(a^3b)/b^2}}}
And the denominator is rational. Next we have to see if the remaining square root can be simplified. For this we look for perfect square factors. We should recognize, that {{{a^2}}} is both a perfect square and it is a factor of the radicand ("radicand" is a word for the expression within a radical).<br>
Now that we have identified a perfect square factor we factor it out:
{{{sqrt(a^2*ab)/b^2}}}
Now we use the first property of radicals to split out the perfect square factor into its own square root:
{{{(sqrt(a^2)*sqrt(ab))/b^2}}}
And finally simplify the square root of the perfect square:
{{{(a*sqrt(ab))/b^2}}}
We have a rational denominator and there are no square roots left which have perfect square factors. This means we are finished.