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

Aug 13, 2017

See a solution process below:

#### Explanation:

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

${\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}}$

${2}^{4} \cdot {\left({2}^{\textcolor{red}{- 1}}\right)}^{\textcolor{b l u e}{4}} \implies {2}^{4} \cdot {2}^{\textcolor{red}{- 1} \times \textcolor{b l u e}{4}} \implies {2}^{4} \cdot {2}^{-} 4$

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

${x}^{\textcolor{red}{a}} = \frac{1}{x} ^ \textcolor{red}{- a}$

${2}^{4} \cdot {2}^{\textcolor{red}{- 4}} \implies {2}^{4} \cdot \frac{1}{2} ^ \textcolor{red}{- - 4} \implies {2}^{4} \cdot \frac{1}{2} ^ \textcolor{red}{4} \implies {2}^{4} / {2}^{4}$

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

${2}^{4} / {2}^{4} \implies \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{{2}^{4}}}}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{{2}^{4}}}}} \implies 1$

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

${x}^{\textcolor{red}{a}} / {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} - \textcolor{b l u e}{b}}$ and ${a}^{\textcolor{red}{0}} = 1$

${2}^{\textcolor{red}{4}} / {2}^{\textcolor{b l u e}{4}} \implies {2}^{\textcolor{red}{4} - \textcolor{b l u e}{4}} \implies {2}^{\textcolor{red}{0}} \implies 1$