document.write( "Question 405426: square root 9x^2y square root3x^5y^2\r
\n" ); document.write( "\n" ); document.write( "i am stuck. I got the obvious the square root of 9 which is 3 but I don't know what to do next and what do to with my variables. My professor ran out of time to go over this in class. please help! \n" ); document.write( "

Algebra.Com's Answer #286464 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"sqrt%289x%5E2y%29%2Asqrt%283x%5E5y%5E2%29\"$
\n" ); document.write( "You are correct in that the square root of the 9 will become a 3 at some point. This can be the first thing you do or it can come later. Since you did this first then that is how I will do it too:
\n" ); document.write( " $\"sqrt%289%29%2Asqrt%28x%5E2y%29%2Asqrt%283x%5E5y%5E2%29\"$
\n" ); document.write( " $\"3%2Asqrt%28x%5E2y%29%2Asqrt%283x%5E5y%5E2%29\"$

\n" ); document.write( "The next thing I would do is multiply the remaining square roots together using the property of radicals, $\"root%28a%2C+p%29%2Aroot%28a%2C+q%29+=+root%28a%2C+p%2Aq%29\"$ :
\n" ); document.write( " $\"3%2Asqrt%283x%5E7y%5E3%29\"$
\n" ); document.write( "Next we look for perfect square factors (other than 1) in $\"3x%5E7y%5E3\"$ . There are no perfect square factors in 3. For the $\"x%5E7\"$ and the $\"y%5E3\"$ you are looking for exponents that are multiples of 2, not perfect squares! For example, $\"a%5E6\"$ , $\"b%5E24\"$ and $\"c%5E208\"$ are all perfect squares not because their exponents are perfect squares (which they are not) but because the exponents are all divisible by 2!

\n" ); document.write( "So $\"x%5E7\"$ and $\"y%5E3\"$ are not perfect squares because their exponents are not divisible by 2. But they both have factors that are perfect squares:
\n" ); document.write( " $\"3%2Asqrt%283%2Ax%5E2%2Ax%5E2%2Ax%5E2%2Ax%2Ay%5E2%2Ay%29\"$
\n" ); document.write( "For reasons that will become clear shortly I like to use the Commutative Property to rearrange the order of the factors so that all the perfect square factors are n front:
\n" ); document.write( " $\"3%2Asqrt%28x%5E2%2Ax%5E2%2Ax%5E2%2Ay%5E2%2A3%2Ax%2Ay%29\"$
\n" ); document.write( "Next we use the same property as before, only in the other direction, to split up this square root of a product into a product of square roots. We want all the perfect square factors in their own square roots. The factors that are not perfect squares all go into the same square root:
\n" ); document.write( "

\n" ); document.write( "The square roots of the perfect squares will all simplify:
\n" ); document.write( " $\"3%2Ax%2Ax%2Ax%2Ay%2Asqrt%283%2Ax%2Ay%29\"$
\n" ); document.write( "or
\n" ); document.write( " $\"3%2Ax%5E3%2Ay%2Asqrt%283%2Ax%2Ay%29\"$
\n" ); document.write( "This is the simplified answer. (Note how the square root is at the end. This is the normal way to write terms like this and it is the reason I put all the perfect square factors in front earlier.) \n" ); document.write( "

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