Question 418900: simplify the fourth root of 2/3
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
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.
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  which is 27:
which simplifies to
Next we use a property of radicals,  , separate the the numerator and denominator into their own 4th roots:
The denominator, since we know 81 is the product of 4 3's, simplifies:
There are no 4th power factors in 54 so we are finished simplifying.
Note: If the expression had been  then we could but would not want to multiply the numerator nd denominator by  . 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
If we had used we would have ended up with
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  ! 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.
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