Questions: Simplify the following expression. Write the result using positive exponents only. Assume that all bases are not equal to 0 . r / (r^(-2) r^(-7)) r / (r^(-2) r^(-7))=

Solution Steps

To simplify the expression \(\frac{r}{r^{-2} r^{-7}}\), we need to apply the laws of exponents. Specifically, we will use the rule that states \(a^{-m} = \frac{1}{a^m}\) and the rule for dividing powers with the same base, which is \(a^m / a^n = a^{m-n}\). First, simplify the expression in the denominator by adding the exponents, then simplify the entire fraction by subtracting the exponents.

We start with the expression:

\[ \frac{r}{r^{-2} r^{-7}} \]

In the denominator, we can combine the exponents:

\[ r^{-2} r^{-7} = r^{-2 + (-7)} = r^{-9} \]

Now, we can rewrite the original expression using the simplified denominator:

\[ \frac{r}{r^{-9}} \]

Using the rule for dividing powers with the same base, we have:

\[ \frac{r^1}{r^{-9}} = r^{1 - (-9)} = r^{1 + 9} = r^{10} \]

Final Answer