SOLUTION: Simplify ^4(over the square root of)243x^8y^10 by factoring while assuming that all expressions under the radical represent nonnegative numbers.

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Question 401457: Simplify ^4(over the square root of)243x^8y^10 by factoring while assuming that all expressions under the radical represent nonnegative numbers.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
root%284%2C+243x%5E8y%5E10%29
First, these are called fourth (4th) roots. The phrase "square root" has nothing to do with this.

What you are calling a square root is actually the radical symbol. The radicals symbol is used for all types of roots. The little number in the upper left corner of the radical is called the index. The index indicates which type of root the radical represents. The index of 4 on your radical tells us that it represents a 4th root. A radical without a visible index has an implied index of 2 and represents a square (or 2nd) root. This is why radicals and square roots can be easily confused.

We can simplify this radical if we can find perfect powers of 4 in the radicand (the expression within a radical). For 243x%5E8y%5E10%29 this means
  • looking for perfect powers of 4 as factors of 243; and/or
  • finding perfect powers of 4 as factors of x%5E8 and y%5E10. Because of the way exponents work, any exponent that is a multiple of 4 is a perfect power of 4.

81 is a perfect power of 4 (because it is 3%5E4) and 81 is a factor of 243. x%5E8 is itself a power of 4 not because 8 is a perfect power of 4 (which it is not) but because 8 is a multiple of 4. y%5E10 is not itself a power of 4 (because the exponent is not divisible by 4). But y%5E10+=+y%5E8%2Ay%5E2 and the y%5E8 is a power of 4. So we can factor the radicand as follows:
root%284%2C+81%2A3%2Ax%5E8%2Ay%5E8%2Ay%5E2%29
For reasons you will see shortly, I like to use the Commutative Property to rearrange the order of the factors so that the perfect power of 4 factors are in front:
root%284%2C+81%2Ax%5E8%2Ay%5E8%2A3%2Ay%5E2%29
Now we use the same property as earlier. But this time we are using it in the opposite direction to take a single 4th root of a product and break it into a product of the 4th roots of each factor:
root%284%2C+81%29%2Aroot%284%2C+x%5E8%29%2Aroot%284%2C+y%5E8%29%2Aroot%284%2C+3%2Ay%5E2%29
Note: The factors that are not powers of 4 are not split into separate radicals because they will not simplify. The 4th roots of the perfect powers of 4 will simplify:

3%2Ax%5E2%2Ay%5E2%2Aroot%284%2C+3%2Ay%5E2%29
Note how the radical is at the end. This is the normal way to write terms like this. The radical is at the end now because we put the perfect power of 4 factors in front earlier.

Note: Even-numbered roots are supposed to be non-negative (unless there is a minus sign in front of them). Therefore when simplifying even-numbered roots you must ensure that the simplified expression is just as non-negative as the original expression. Absolute values are often used to ensure non-negative expressions. We did not use any abolsute values in this problem for either of the following two reasons:
  • We were told that all variables were non-negative; or
  • Our simplified 4th roots could not be negative:
    • 3 is clearly not negative.
    • x%5E2 and y%5E2 could not be negative because you can't get a negative number when squaring a (real) number.