document.write( "Question 352392: Use rational exponents to simplify. Write the answer in radical notation, if possible.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "exponent of 12 SQRT (a^10) \n" ); document.write( "

Algebra.Com's Answer #251939 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
For this problem you need to understand:

The relationship between radicals and rational/fractional exponents: $\"q%5E%28a%2Fb%29+=+root%28b%2C+q%5Ea%29+=+%28root%28b%2C+q%29%29%5Ea\"$
Square roots are \"second roots\". IOW: $\"sqrt%28x%29\"$ is the same as $\"root%282%2C+x%29\"$ .

\n" ); document.write( "So $\"12sqrt%28a%5E10%29\"$ written with rational exponents would be:
\n" ); document.write( " $\"12%2A%28a%5E10%29%5E%281%2F2%29\"$
\n" ); document.write( "Using the rule for exponents, $\"%28a%5Ep%29%5Eq+=+a%5E%28%28p%2Aq%29%29\"$ , we can simplify the \"a\" part of the above:
\n" ); document.write( " $\"12%2Aa%5E%28%2810%2A%281%2F2%29%29%29\"$
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"12a%5E5\"$

\n" ); document.write( "There is only one thing left. All square roots are supposed to be positive or zero, including $\"sqrt%28a%5E10%29\"$ . And 12 times a positive number or zero, like 12sqrt(a^10), will be positive or zero. However, our answer at this point is not necessarily positive or zero. In fact, if \"a\" is negative then $\"12a%5E5\"$ will be negative, too. Since \"a\" could be negative (there's no reason it can't be), we cannot use $\"12a%5E5\"$ for the answer. We must somehow ensure that our answer must be positive or zero. There are a variety of expressions we could use. Probably the preferred one is found as follows:
\n" ); document.write( " $\"12a%5E5\"$
\n" ); document.write( " $\"12a%5E4%2Aa\"$
\n" ); document.write( " $\"12a%5E4%2Aabs%28a%29\"$
\n" ); document.write( "This expression will always be positive or zero no matter what \"a\" is. \n" ); document.write( "

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