document.write( "Question 398745: simplify (25/x^-4)^(3/2) \n" ); document.write( "

Algebra.Com's Answer #282460 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"%2825%2Fx%5E%28-4%29%29%5E%283%2F2%29\"$
\n" ); document.write( "First let's work within the parentheses. In general $\"a%5E%28-n%29+=+1%2Fa%5En\"$ and $\"1%2Fa%5E%28-n%29+=+a%5En\"$ . Using the second pattern, the expression within the parentheses becomes:
\n" ); document.write( " $\"%2825x%5E4%29%5E%283%2F2%29\"$
\n" ); document.write( "Now we can use a rule for exponents, $\"%28ab%29%5En+=+a%5En%2Ab%5En\"$ , to raise $\"25x%5E4\"$ to the 3/2 power:
\n" ); document.write( " $\"25%5E%283%2F2%29%2A%28x%5E4%29%5E%283%2F2%29\"$
\n" ); document.write( "Simplifying the last part of the expression is simple. The rule for exponents when raising a power to a power is to multiply the exponents:
\n" ); document.write( " $\"25%5E%283%2F2%29%2Ax%5E%284%2A%283%2F2%29%29\"$
\n" ); document.write( " $\"25%5E%283%2F2%29%2Ax%5E6\"$
\n" ); document.write( "Simplifying the first part can be difficult for those who do not yet understand fractional exponents. I find that it can help if you factor the exponent in a special way:

If the exponent is negative, factor out -1.
If the exponent is fractional and the numerator is not a 1, then factor out the numerator.

\n" ); document.write( "The exponent on the 25 is not negative. But it is fractional and its denominator is not a 1. So we will factor out the 3:
\n" ); document.write( " $\"25%5E%283%2A%281%2F2%29%29%2Ax%5E6\"$
\n" ); document.write( "Each of the factors tells us an operation to perform. The 3 tells us that we will be cubing. And the 1/2, if you remember what fractional exponents mean, tells us that we will be finding a square root. And since multiplication is Commutative, we can do these operations in any order we choose. Cubing 25 doesn't look very appealing while finding a square root of 25 looks very easy. So we will start with the square root:
\n" ); document.write( " $\"25%5E%28%281%2F2%29%2A3%29%2Ax%5E6\"$
\n" ); document.write( " $\"%2825%5E%281%2F2%29%29%5E3%2Ax%5E6\"$
\n" ); document.write( " $\"%285%29%5E3%2Ax%5E6\"$
\n" ); document.write( " $\"125x%5E6\"$

\n" ); document.write( "Note: If we had chosen to cube 25 first and then find a square root, we would still have gotten 125 -- just not as easily. \n" ); document.write( "

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