Simplify. Express your answer as a single term, without a denominator. q^0 * r^(-1) * s^1 * q^r * s^(-1) * r^8

Steps to Solve

1. Use the property of exponents
   The expression includes terms with exponents. Recall the property that states $a^0 = 1$ for any non-zero number $a$. Thus, $q^0 = 1$.
   So the expression simplifies to:
   $$1 \cdot r^{-1} \cdot s^1 \cdot q^r \cdot s^{-1} \cdot r^8$$
   This reduces to:
   $$r^{-1} \cdot s \cdot q^r \cdot s^{-1} \cdot r^8$$

2. Combine terms with the same base
   Next, simplify the expression further by combining like terms. Notice that $s \cdot s^{-1} = s^{1-1} = s^0 = 1$. This means:
   $$r^{-1} \cdot 1 \cdot q^r \cdot r^8$$
   Which simplifies to:
   $$r^{-1} \cdot q^r \cdot r^8$$

3. Add or subtract the exponents for like bases
   Now, combine the terms with base $r$. The exponents are $-1$ and $8$, so add them together:
   $$r^{-1 + 8} = r^{7}$$
   Thus, we have:
   $$r^7 \cdot q^r$$

4. Express as a single term without a denominator
   Finally, write the expression as:
   $$r^{7} q^r$$