SOLUTION: i must simplify the suare root of a/b times the square root of a^2/b^2 thanks for the help

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Question 277473: i must simplify the suare root of a/b times the square root of a^2/b^2 thanks for the help
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%28a%2Fb%29%2Asqrt%28a%5E2%2Fb%5E2%29
When simplifying expressions of square roots there are two issues to address:
  • Simplifying the square roots by extracting perfect square factors, if any
  • Making sure any denominators are rational (i.e. no square roots in denominators.

To accomplish these tasks we often need one or both of the following properties of radicals:
  • root%28a%2C+p%29+%2A+root%28a%2C+q%29+=+root%28a%2C+p%2Aq%29
  • root%28a%2C+p%29+%2F+root%28a%2C+q%29+=+root%28a%2C+p%2Fq%29

These properties can and often are used in both "directions". For example the first property lets us change
sqrt%28ab%5E2%29 into sqrt%28a%29+%2A+sqrt%28b%5E2%29
It also lets us change
sqrt%288%29%2Asqrt%282%29 into sqrt%288%2A2%29

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:
%28sqrt%28a%29%2Fsqrt%28b%29%29%2A%28sqrt%28a%5E2%29%2Fsqrt%28b%5E2%29%29
Now we'll multiply the two fractions using the first property to multiply the square roots:
sqrt%28a%5E3%29%2Fsqrt%28b%5E3%29
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%28b%5E3%29. 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%28b%29:
%28sqrt%28a%5E3%29%2Fsqrt%28b%5E3%29%29%28sqrt%28b%29%2Fsqrt%28b%29%29
which results in:
sqrt%28a%5E3b%29%2Fsqrt%28b%5E4%29
The denominator simplifies to:
sqrt%28a%5E3b%29%2Fb%5E2
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%5E2 is both a perfect square and it is a factor of the radicand ("radicand" is a word for the expression within a radical).

Now that we have identified a perfect square factor we factor it out:
sqrt%28a%5E2%2Aab%29%2Fb%5E2
Now we use the first property of radicals to split out the perfect square factor into its own square root:
%28sqrt%28a%5E2%29%2Asqrt%28ab%29%29%2Fb%5E2
And finally simplify the square root of the perfect square:
%28a%2Asqrt%28ab%29%29%2Fb%5E2
We have a rational denominator and there are no square roots left which have perfect square factors. This means we are finished.