# Question #70369

Jun 17, 2017

See a solution process below:

#### Explanation:

We can use this rule for exponents to rewrite this expression:

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

Substituting the exponents from the problem gives:

${x}^{\textcolor{red}{1 - a}} / {x}^{\textcolor{b l u e}{a}} = {x}^{\textcolor{red}{1 - a} - \textcolor{b l u e}{a}} = {x}^{\textcolor{red}{1 - 1 a} - \textcolor{b l u e}{1 a}} = {x}^{1 - 2 a}$

Depending on the value of $a$ this may result in a negative exponent. If we want an answer with just positive exponents we can also rewrite this expression using this rule of exponents:

${x}^{\textcolor{red}{a}} / {x}^{\textcolor{b l u e}{b}} = \frac{1}{x} ^ \left(\textcolor{b l u e}{b} - \textcolor{red}{a}\right)$

Again, substituting the exponents from the problem gives:

${x}^{\textcolor{red}{1 - a}} / {x}^{\textcolor{b l u e}{a}} = \frac{1}{x} ^ \left(\textcolor{b l u e}{a} - \textcolor{red}{\left(1 - a\right)}\right) \implies \frac{1}{x} ^ \left(\textcolor{b l u e}{a} - \textcolor{red}{1 + a}\right) \implies \frac{1}{x} ^ \left(\textcolor{b l u e}{1 a} - \textcolor{red}{1 + 1 a}\right) \implies \frac{1}{x} ^ \left(2 a - 1\right)$