document.write( "Question 277290: can you solve this problem as an example to my notes for class please?
\n" ); document.write( "the square root of 8x divided by the square root of 3x to the third power \n" ); document.write( "

Algebra.Com's Answer #201972 by jsmallt9(3759) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"sqrt%288x%29%2Fsqrt%283x%5E3%29\"$
\n" ); document.write( "There are two main issues when simplifying square roots:

Finding and extracting prefect square factors, if any
Making any denominators rational (i.e. No irrational numbers like square roots in denominators.)

\n" ); document.write( "These issues can be addressed in any order. Generally I like to address the rational denominators, first. Sometimes, if you extract the perfect square factors first and then rationalize the denominators, you end up having to extract perfect square factors again. (When you reduce fractions, you keep reducing until you can't reduce them any further. Extracting perfect square factors is like reducing fractions in the sense that you keeping going until you can't go any further.)

\n" ); document.write( "So I'll start with rationalizing the denominator. With a monomial (one-term expression) like this you rationalize the denominator by multiplying the numerator and denominator by whatever it takes to turn the denominator into a perfect square. With a denominator of $\"sqrt%283x%5E3%29\"$ the obvious choice is to multiply by $\"sqrt%283x%5E3%29\"$ . And it would not be wrong to do this. But there is a better choice. The smallest achievable perfect square is best. And with this denominator we can multiply by $\"sqrt%283x%29\"$ and get a perfect square. So this is what I will use. (If you use $\"sqrt%283x%5E3%29\"$ you will still get the answer right but it there will more work to do. It's a little like the using a denominator of 8 when adding 1/2 and 1/4.)
\n" ); document.write( " $\"%28sqrt%288x%29%2Fsqrt%283x%5E3%29%29%28sqrt%283x%29%2Fsqrt%283x%29%29\"$
\n" ); document.write( "Multiplying we get:
\n" ); document.write( " $\"sqrt%2824x%5E2%29%2Fsqrt%28%283x%5E3%29%5E2%29\"$
\n" ); document.write( "and the denominator simplifies:
\n" ); document.write( " $\"sqrt%2824x%5E2%29%2F%283x%5E3%29\"$
\n" ); document.write( "and the square root in the denominator is gone. Now we turn to extracting perfect square factors, if any, from the remaining square root:
\n" ); document.write( " $\"sqrt%284%2A6%2Ax%5E2%29%2F%283x%5E3%29\"$
\n" ); document.write( " $\"%28sqrt%284%29%2Asqrt%286%29%2Asqrt%28x%5E2%29%29%2F%283x%5E3%29\"$
\n" ); document.write( "Replacing the square roots of the perfect squares we get:
\n" ); document.write( " $\"%282%2Asqrt%286%29%2Ax%29%2F%283x%5E3%29\"$
\n" ); document.write( "The x in the numerator cancels with one of the three factors of x in the denominator leaving:
\n" ); document.write( " $\"%282%2Asqrt%286%29%29%2F%283x%5E2%29\"$ \n" ); document.write( "

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