Let's break down the steps of simplifying the given expression [tex]\( r^{-8} s^{-5} \)[/tex] and then identify the error that Owen might have made.
### Simplification Steps:
1. Understanding Negative Exponents:
- If an expression has a negative exponent, [tex]\( a^{-n} \)[/tex], it can be rewritten as [tex]\( \frac{1}{a^n} \)[/tex]. This is the fundamental rule for dealing with negative exponents.
2. Applying the Rule to Each Factor Separately:
- For [tex]\( r^{-8} \)[/tex]:
[tex]\[
r^{-8} = \frac{1}{r^8}
\][/tex]
- For [tex]\( s^{-5} \)[/tex]:
[tex]\[
s^{-5} = \frac{1}{s^5}
\][/tex]
3. Combining the Results:
- Since the original expression is [tex]\( r^{-8} s^{-5} \)[/tex], and we have rewritten each part, we can combine them:
[tex]\[
r^{-8} s^{-5} = \left(\frac{1}{r^8}\right) \cdot \left(\frac{1}{s^5}\right)
\][/tex]
4. Multiplying the Fractions:
- When we multiply two fractions, we multiply the numerators and the denominators respectively:
[tex]\[
\left(\frac{1}{r^8}\right) \cdot \left(\frac{1}{s^5}\right) = \frac{1 \cdot 1}{r^8 \cdot s^5} = \frac{1}{r^8 s^5}
\][/tex]
### Simplified Expression:
[tex]\[
r^{-8} s^{-5} = \frac{1}{r^8 s^5}
\][/tex]
### Identifying Owen's Error:
Owen seems to have misunderstood the rule for negative exponents. According to the description given:
[tex]\[
s_s=\frac{1}{s^s} \cdot s^s=\frac{s^s}{s^s}
\][/tex]
This indicates that Owen mistakenly believes that multiplying the fractions and then reciprocating will result in the same expression being canceled out, which is incorrect. The concept of negative exponents should lead directly to the transformation as described above and not result in canceling out terms or misapplying exponent rules in such a manner.
### Corrected Concept:
The correct approach should recognize that:
1. Each negative exponent results in a term moving to the denominator.
2. Multiplying such fractions directly to form a single simplified fraction.
Therefore, the correct simplified expression is:
[tex]\[
r^{-8} s^{-5} = \frac{1}{r^8 s^5}
\][/tex]
This explanation demonstrates the correct use of negative exponent rules and where Owen's error occurred in his simplification process.
