document.write( "Question 277473: i must simplify the suare root of a/b times the square root of a^2/b^2 thanks for the help \n" ); document.write( "

Algebra.Com's Answer #202076 by jsmallt9(3758) $\"\"$ $\"About$

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$\"sqrt%28a%2Fb%29%2Asqrt%28a%5E2%2Fb%5E2%29\"$
\n" ); document.write( "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.

\n" ); document.write( "To accomplish these tasks we often need one or both of the following properties of radicals:

$\"root%28a%2C+p%29+%2A+root%28a%2C+q%29+=+root%28a%2C+p%2Aq%29\"$
$\"root%28a%2C+p%29+%2F+root%28a%2C+q%29+=+root%28a%2C+p%2Fq%29\"$

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

\n" ); document.write( "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:
\n" ); document.write( "Use the second property to split the square roots of the fractions into fractions of square roots:
\n" ); document.write( " $\"%28sqrt%28a%29%2Fsqrt%28b%29%29%2A%28sqrt%28a%5E2%29%2Fsqrt%28b%5E2%29%29\"$
\n" ); document.write( "Now we'll multiply the two fractions using the first property to multiply the square roots:
\n" ); document.write( " $\"sqrt%28a%5E3%29%2Fsqrt%28b%5E3%29\"$
\n" ); document.write( "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 $\"sqrt%28b%29\"$ :
\n" ); document.write( " $\"%28sqrt%28a%5E3%29%2Fsqrt%28b%5E3%29%29%28sqrt%28b%29%2Fsqrt%28b%29%29\"$
\n" ); document.write( "which results in:
\n" ); document.write( " $\"sqrt%28a%5E3b%29%2Fsqrt%28b%5E4%29\"$
\n" ); document.write( "The denominator simplifies to:
\n" ); document.write( " $\"sqrt%28a%5E3b%29%2Fb%5E2\"$
\n" ); document.write( "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).

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

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