document.write( "Question 374185: How would you evaluate an expression with a mixed-number exponent such as 8^1 1/3? \n" ); document.write( "

Algebra.Com's Answer #266298 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"8%5E%281%261%2F3%29\"$
\n" ); document.write( "Mixed numbers are almost always a pain. I usually recommend that they be converted to improper fractions. It is the same here. So we will start by changing $\"1%261%2F3\"$ to 4/3:
\n" ); document.write( " $\"8%5E%284%2F3%29\"$
\n" ); document.write( "If you are still not sure how to simplify this, I recommend writing the exponent in a factored form. In this case, we would rewrite the expression as:
\n" ); document.write( " $\"8%5E%284%2A%281%2F3%29%29\"$
\n" ); document.write( "Since
\n" ); document.write( " $\"8%5E%284%2A%281%2F3%29%29+=+%288%5E4%29%5E%281%2F3%29\"$
\n" ); document.write( "which says that we raise 8 to the 4th power and then, because an exponent of 1/3 means cube root, find the cube root of the answer.
\n" ); document.write( "Also, since multiplication is Commutative, we are free to change the order:
\n" ); document.write( " $\"8%5E%28%281%2F3%29%2A4%29+=+%288%5E%281%2F3%29%29%5E4\"$
\n" ); document.write( "which says to find a cube root and then raise to the 4th power.

\n" ); document.write( "So, in summary, an exponent of $\"4%2A%281%2F3%29\"$ means we will raise to the 4th power and we will find a cube root and we get to choose the order in which these are done!

\n" ); document.write( "Since 8 is a perfect cube ( $\"8+=+2%5E3\"$ ), it seems to me that finding the cube root first will be easier than raising 8 to the 4th power. So here's the problem from start to finish:
\n" ); document.write( "

\n" ); document.write( "(Note: even if you decide to raise to the 4th power first and then find the cube root, you still end up with 16.) \n" ); document.write( "

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