Let's go through the detailed, step-by-step solution to identify Tyler's error and correctly simplify the expression [tex]\( x^{-3} y^{-9} \)[/tex].
1. Original Expression:
[tex]\[
x^{-3} y^{-9}
\][/tex]
2. Applying the Property of Negative Exponents:
Recall that for any non-zero number [tex]\( a \)[/tex] and an integer [tex]\( n \)[/tex]:
[tex]\[
a^{-n} = \frac{1}{a^n}
\][/tex]
Using this property on [tex]\( x^{-3} \)[/tex] and [tex]\( y^{-9} \)[/tex], we get:
[tex]\[
x^{-3} = \frac{1}{x^3}
\][/tex]
[tex]\[
y^{-9} = \frac{1}{y^9}
\][/tex]
3. Simplifying the Expression:
Substitute the simplified forms back into the original expression:
[tex]\[
x^{-3} y^{-9} = \frac{1}{x^3} \cdot \frac{1}{y^9}
\][/tex]
4. Combining the Fractions:
Using the property of fractions [tex]\(\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\)[/tex], we can combine the fractions:
[tex]\[
\frac{1}{x^3} \cdot \frac{1}{y^9} = \frac{1 \cdot 1}{x^3 \cdot y^9} = \frac{1}{x^3 y^9}
\][/tex]
5. Final Simplified Expression:
[tex]\[
x^{-3} y^{-9} = \frac{1}{x^3 y^9}
\][/tex]
6. Identifying Tyler's Error:
Tyler's simplification steps were:
[tex]\[
x^{-3} y^{-9} \rightarrow \frac{1}{x^3} \cdot \frac{1}{y^{-9}} = \frac{1}{x^3 y^{-9}}
\][/tex]
The mistake here is in not correctly applying the property of negative exponents to [tex]\( y^{-9} \)[/tex]. The [tex]\( y^{-9} \)[/tex] term should be in the denominator, with a positive exponent.
Therefore, the correct version should be:
[tex]\[
\frac{1}{x^3} \cdot \frac{1}{y^9} = \frac{1}{x^3 y^9}
\][/tex]
Correct Answer:
The power [tex]\( y^9 \)[/tex] should be in the numerator and the power [tex]\( x^3 \)[/tex] in the denominator.
Therefore, the correct identification of Tyler's error is:
"The power [tex]\( y^9 \)[/tex] should be in the numerator and the power [tex]\( x^3 \)[/tex] in the denominator."
