document.write( "Question 398497: simplify by factoring the square root of 75a^2b
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #282514 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"sqrt%2875a%5E2b%29\"$
\n" ); document.write( "When they say \"simplify [a square root] by factoring\" they mean \"Simplify by finding perfect square factors (other than 1)\". So we want to factor $\"75a%5E2b\"$ into as many prefect square factors (other than 1) as we can.

\n" ); document.write( "Clearly $\"a%5E2\"$ is a perfect square factor. We can also factor 75 into 25*3:
\n" ); document.write( " $\"sqrt%2825%2A3%2Aa%5E2%2Ab%29\"$
\n" ); document.write( "For reasons that will become clear shortly, I like to use the Commutative Property at this point to rearrange the order of the multiplication so that the perfect square factors are in front:
\n" ); document.write( " $\"sqrt%2825%2Aa%5E2%2A3%2Ab%29\"$
\n" ); document.write( "Now we use a property of all radicals (not just square roots), $\"root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%28a%2C+q%29\"$ , to split the square root of the product into the product of the square roots of the factors. Each perfect square factor goes into its own square root. The \"non-prefect square\" factors all go into one square root:
\n" ); document.write( " $\"sqrt%2825%29%2Asqrt%28a%5E2%29%2Asqrt%283%2Ab%29\"$
\n" ); document.write( "The square roots of the perfect squares simplify. Since square roots are supposed to be positive and since we do not know if \"a\" is positive, we cannot just say that $\"sqrt%28a%5E2%29+=+a\"$ as you might think. We must use absolute value to ensure that $\"sqrt%28a%5E2%29\"$ remains positive:
\n" ); document.write( " $\"5%2Aabs%28a%29%2Asqrt%283b%29\"$
\n" ); document.write( "This is the simplified expression. (Note how the square root is at the end of this expression. This is the usual way of writing a term that includes a square root. And this is why I put the perfect square factors at the front earlier.)

\n" ); document.write( "Note: Often problems like this include some statement that says that all variables should be considered to be positive. If your problem says something like this (you didn't mention it), then the absolute value is not necessary and you end up with: $\"5a%2Asqrt%283b%29\"$ \n" ); document.write( "

\n" );