document.write( "Question 352453: Use rational exponents to write an exponent of 4 SQRT(x) * an exponent of 3 SQRT(6x) as a single radical expression??
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Algebra.Com's Answer #251932 by jsmallt9(3758)\"\" \"About 
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I assume the expression is:
\n" ); document.write( "\"root%284%2C+x%29%2Aroot%283%2C+6x%29\"
\n" ); document.write( "If this is correct, the first radical is read: \"the fourth root of x\" and the second radical is read: \"the third root of 6x\" or \"the cube root of 6x\". The 4 and the 3 are not exponents (although, as you'll see shortly, they do become denominators of fractional exponents).

\n" ); document.write( "Another word for \"rational\" (in Math) is \"fractional\". So the problem is saying: \"Use fractional exponents to ...\" This means we need to know how to change radicals into fractional exponents (and vice versa). The connection between radicals and exponents is:
\n" ); document.write( "\"q%5E%28a%2Fb%29+=+root%28b%2C+q%5Ea%29+=+%28root%28b%2C+q%29%29%5Ea\"
\n" ); document.write( "(Note: square roots are \"second roots\". IOW, \"sqrt%28x%29\" is the same thing as \"root%282%2C+x%29\".)

\n" ); document.write( "So your expression, written with fractional exponents instead of radicals is:
\n" ); document.write( "\"x%5E%281%2F4%29%2A%286x%29%5E%281%2F3%29\"
\n" ); document.write( "(Note: Are you sure the 6 is in the problem? It makes the problem much harder.) Using the the rule for exponents, \"%28a%2Ab%29%5Eq+=+a%5Eq%2Ab%5Eq\", on the 6x part of the above expression we get:
\n" ); document.write( "\"x%5E%281%2F4%29%2A6%5E%281%2F3%29x%5E%281%2F3%29\"
\n" ); document.write( "To multiply the \"x\" parts of the above we will use a rule for exponents, \"q%5Ea%2Aq%5Eb+=+q%5E%28a%2Bb%29\", which says we should add the exponents:
\n" ); document.write( "\"6%5E%281%2F3%29x%5E%28%28%281%2F3%29+%2B+%281%2F4%29%29%29\"
\n" ); document.write( "To add the exponents, which are fractions, we must, of course, have common denominators:
\n" ); document.write( "\"6%5E%281%2F3%29x%5E%28%28%284%2F12%29+%2B+%283%2F12%29%29%29\"
\n" ); document.write( "\"6%5E%281%2F3%29x%5E%287%2F12%29\"
\n" ); document.write( "To combine the 6 part and the \"x\" part of the above we will use the \"%28a%2Ab%29%5Eq+=+a%5Eq%2Ab%5Eq\" rule again (in the opposite direction this time). Once again we need a common denominator:
\n" ); document.write( "\"6%5E%284%2F12%29x%5E%287%2F12%29\"
\n" ); document.write( "Factoring out a 1/12 in each exponent we get:
\n" ); document.write( "\"6%5E%28%284%2A%281%2F12%29%29%29x%5E%28%287%2A%281%2F12%29%29%29\"
\n" ); document.write( "Now we can use the \"%28a%2Ab%29%5Eq+=+a%5Eq%2Ab%5Eq\" rule (from right to left this time):
\n" ); document.write( "\"%286%5E4x%5E7%29%5E%281%2F12%29\"
\n" ); document.write( "Now, since 1/12 as an exponent means 12th root, we can rewrite the above as the radical:
\n" ); document.write( "\"root%2812%2C+6%5E4%2Ax%5E7%29\"
\n" ); document.write( "The only thing left is to multiply out \"6%5E4\". I'll leave that to you and your calculator. (It should work out to be a number somewhere near 1300.) \n" ); document.write( "
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