How do you simplify #2^4*(2^-1)^4# and write it using only positive exponents?

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Answer 1 · 2017-08-13T14:37:57

See a solution process below:

First, we can use this rule of exponents to simplify the term on the right side of the expression:

#(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))#

#2^4 * (2^color(red)(-1))^color(blue)(4) => 2^4 * 2^(color(red)(-1) xx color(blue)(4)) => 2^4 * 2^-4#

We can now use this rule of exponents to rewrite the term on the right with positive exponents:

#x^color(red)(a) = 1/x^color(red)(-a)#

#2^4 * 2^color(red)(-4) => 2^4 * 1/2^color(red)(- -4) => 2^4 * 1/2^color(red)(4) => 2^4/2^4#

One way to simplify is to simply cancel to common terms in the numerator and denominator:

#2^4/2^4 => color(red)(cancel(color(black)(2^4)))/color(red)(cancel(color(black)(2^4))) => 1#

Another way is to use these rules of exponents to get to the same solution:

#x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))# and #a^color(red)(0) = 1#

#2^color(red)(4)/2^color(blue)(4) => 2^(color(red)(4)-color(blue)(4)) => 2^color(red)(0) => 1#