document.write( "Question 418900: simplify the fourth root of 2/3 \n" ); document.write( "

Algebra.Com's Answer #293155 by jsmallt9(3758) $\"\"$ $\"About$

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$\"root%284%2C+2%2F3%29\"$
\n" ); document.write( "Part of simplifying radicals is eliminating:

any radicals in denominators, and
any fractions within a radical.

You have a fraction in a radical to eliminate.

\n" ); document.write( "To eliminate a fraction within a radical, I like to multiply the numerator and denominator of that fraction by any number that will turn the denominator into a power that matches the type of root. With this expression, with its 4th root, we are looking to make the denominator a perfect 4th power. It is to your advantage to use the smallest number that will make the denominator a perfect 4th power. In this case, since the denominator is 3, a prime number, the lowest number that will turn a 3 into a perfect 4th power will be $\"3%5E3\"$ which is 27:
\n" ); document.write( " $\"root%284%2C+%282%2F3%29%2827%2F27%29%29\"$
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"root%284%2C+54%2F81%29\"$
\n" ); document.write( "Next we use a property of radicals, $\"root%28a%2C+p%2Fq%29+=+root%28a%2C+p%29%2Froot%28a%2C+q%29\"$ , separate the the numerator and denominator into their own 4th roots:
\n" ); document.write( " $\"root%284%2C+54%29%2Froot%284%2C+81%29\"$
\n" ); document.write( "The denominator, since we know 81 is the product of 4 3's, simplifies:
\n" ); document.write( " $\"root%284%2C+54%29%2F3\"$
\n" ); document.write( "There are no 4th power factors in 54 so we are finished simplifying.

\n" ); document.write( "Note: If the expression had been $\"root%284%2C+3%2F4%29\"$ then we could but would not want to multiply the numerator nd denominator by $\"4%5E3\"$ . This is so because 4 is not prime. 4 = 2*2. So all we need to get a perfect 4th power is two more 2's: 2*2 or 4. So
\n" ); document.write( "

\n" ); document.write( "If we had used $\"4%5E3\"$ we would have ended up with
\n" ); document.write( " $\"root%284%2C+192%29%2F4\"$
\n" ); document.write( "This is correct, too. But it is not fully simplified. The numerator will simplify. And after that the fraction will reduce and (surprise, surprise) we end up with $\"root%284%2C+12%29%2F2\"$ ! So you can see that when you don't turn the denominator into the lowest 4th power possible you end up with significant extra work. \n" ); document.write( "

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