document.write( "Question 274380: (1/243)^-3/5 \n" ); document.write( "
Algebra.Com's Answer #200311 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"%281%2F243%29%5E%28-3%2F5%29\"
\n" ); document.write( "Until you get comfortable with exponents, a \"trick\" you can use to make simplifying expressions like this easier is to factor the exponent into \"pieces\".

\n" ); document.write( "\"-3%2F5\" can be factored into \"-1%2A3%2A%281%2F5%29\". And since multiplication is commutative we can order the factors any way we choose. I will choose: \"-1%2A%281%2F5%29%2A3\" for reasons you will soon see. (Note: The simplification can be done using any order. The order I chose makes the problem a little easier as I hope you'll see.)

\n" ); document.write( "Rewriting the exponent in factored form we get:
\n" ); document.write( "\"%281%2F243%29%5E%28-1%2A%281%2F5%29%2A3%29\"
\n" ); document.write( "Now I'll use the power of a power rule for exponents, \"a%5E%28p%2Aq%29+=+%28a%5Ep%29%5Eq\", to rewrite this as powers of powers:
\n" ); document.write( "\"%28%28%281%2F243%29%5E%28-1%29%29%5E%281%2F5%29%29%5E3%29\"
\n" ); document.write( "Now we can apply the exponents, from the inside out. Since -1 as an exponent means reciprocal the inner expression simplifies to:
\n" ); document.write( "\"%28243%5E%281%2F5%29%29%5E3%29\"
\n" ); document.write( "As you can see, the fraction is gone! This makes the rest of the problem simpler. And this is exactly why I chose to put the -1 factor of the exponent first: To get rid of the fraction ASAP.

\n" ); document.write( "Next we'll apply the 1/5 exponent. 1/5 as an exponent means 5th root. And since \"243+=+3%5E5\", the 5th root of 243 is 3. So now we have:
\n" ); document.write( "\"3%5E3\"
\n" ); document.write( "The 243 is now a 3, a much smaller number which makes the remainder easier. And this is why I put the 1/5 factor second. I knew that the 5th root of any number over 1 is a much smaller number.

\n" ); document.write( "\"3%5E3\" is 27. So your original expression simplifies down to 27. \n" ); document.write( "
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