document.write( "Question 257399: Can someone please help me solve these???
\n" ); document.write( "1.(-16)^1/4
\n" ); document.write( "2. 25^-3/2
\n" ); document.write( "3. (-27x^9)1/3 \n" ); document.write( "

Algebra.Com's Answer #189341 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
When working with fractional exponents you need to know about the properties of exponents:

$\"a%5Ep+%2A+a%5Eq+=+a%5E%28p%2Bq%29\"$
$\"a%5Ep+%2F+a%5Eq+=+a%5E%28p-q%29\"$
$\"%28a%5Ep%29%5Eq+=+a%5E%28p%2Aq%29\"$
$\"%28ab%29%5Ep+=+a%5Ep%2Ab%5Ep\"$
$\"a%5E%28-p%29+=+1%2Fa%5Ep\"$
$\"a%5E%28p%2Fq%29+=+root%28q%2C+a%5Ep%29+=+%28root%28q%2C+a%29%29%5Ep\"$

\n" ); document.write( "Let's see how these properties help us simplify your expressions:
\n" ); document.write( "1. $\"%28-16%29%5E%281%2F4%29\"$
\n" ); document.write( "By property #6
\n" ); document.write( " $\"%28-16%29%5E%281%2F4%29+=+root%284%2C+%28-16%29%5E1%29+=+root%284%2C+-16%29\"$
\n" ); document.write( "This expression represents the 4th root of -16. In other words the number which, when raised to the 4th power, results in -16. There are no such numbers in the set of Real numbers. Whenever we raise a Real number to the 4th power we always get a positive result. If you have never heard of Complex Numbers then we can go no further so skip to #2.

\n" ); document.write( "If you know about Complex Numbers, Complex Numbers in Polar form and finding roots of Complex Numbers then we can come up with an answer. Writing $\"%28-16%29%5E%281%2F4%29\"$ as a Complex Number in Polar form we get:
\n" ); document.write( " $\"%2816%28cos%28pi%29%2Bi%2Asin%28pi%29%29%29%5E%281%2F4%29\"$
\n" ); document.write( "Using deMoivre's (spelling?) Theorem this is equal to:
\n" ); document.write( " $\"16%5E%281%2F4%29%28cos%28%281%2F4%29pi%29%2Bi%2Asin%28%281%2F4%29pi%29%29\"$
\n" ); document.write( "Which simplifies as follows:
\n" ); document.write( " $\"%282%29%28sqrt%282%29%2F2%2Bi%2Asqrt%282%29%2F2%29\"$
\n" ); document.write( " $\"sqrt%282%29%2Bi%2Asqrt%282%29%29%29\"$ This is the primary 4th root. The other 3 are:
\n" ); document.write( " $\"sqrt%282%29-i%2Asqrt%282%29%29%29\"$
\n" ); document.write( " $\"-sqrt%282%29%2Bi%2Asqrt%282%29%29%29\"$
\n" ); document.write( " $\"-sqrt%282%29-i%2Asqrt%282%29%29%29\"$
\n" ); document.write( "

\n" ); document.write( "2. $\"25%5E%28-3%2F2%29\"$
\n" ); document.write( "By property #5 above this is equal to:
\n" ); document.write( " $\"1%2F25%5E%283%2F2%29\"$
\n" ); document.write( "By property #6 the denominator can be written as $\"sqrt%2825%5E3%29\"$ or $\"%28sqrt%2825%29%29%5E3\"$ . Since the second form looks easier (after all we do know what the square root of 25 is) we will use that form:
\n" ); document.write( " $\"1%2F%28sqrt%2825%29%29%5E3+=+1%2F%285%29%5E3+=+1%2F125\"$

\n" ); document.write( "3. $\"%28-27x%5E9%29%5E%281%2F3%29\"$
\n" ); document.write( "Using property #4 above we get:
\n" ); document.write( " $\"%28-27%29%5E%281%2F3%29%2A%28x%5E9%29%5E%281%2F3%29\"$
\n" ); document.write( "Using property #6 on the first part and property #3 on the second part we get:
\n" ); document.write( " $\"root%283%2C+-27%29%2Ax%5E%289%2A%281%2F3%29%29\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"-3x%5E3\"$ \n" ); document.write( "

\n" );