To simplify the expression
[tex]\[
\frac{x^{-9} y^{-8} z^{-2}}{x^5 y^7 z^3}
\][/tex]
we need to handle the exponents by using the rules of exponents, specifically the rule that states \( a^m \div a^n = a^{m-n} \).
First, let's look at the exponents for each base separately:
1. Base \( x \):
[tex]\[
\frac{x^{-9}}{x^5} = x^{-9 - 5} = x^{-14}
\][/tex]
2. Base \( y \):
[tex]\[
\frac{y^{-8}}{y^7} = y^{-8 - 7} = y^{-15}
\][/tex]
3. Base \( z \):
[tex]\[
\frac{z^{-2}}{z^3} = z^{-2 - 3} = z^{-5}
\][/tex]
Now, we combine these results:
[tex]\[
x^{-14} y^{-15} z^{-5}
\][/tex]
Following the property of exponents that \( a^{-n} = \frac{1}{a^n} \), we convert each term with a negative exponent to its reciprocal form:
[tex]\[
x^{-14} = \frac{1}{x^{14}}, \quad y^{-15} = \frac{1}{y^{15}}, \quad z^{-5} = \frac{1}{z^5}
\][/tex]
Putting these together, the expression becomes:
[tex]\[
\frac{1}{x^{14}} \cdot \frac{1}{y^{15}} \cdot \frac{1}{z^5} = \frac{1}{x^{14} y^{15} z^5}
\][/tex]
Therefore, the simplified form of the expression using only positive exponents is:
[tex]\[
\frac{1}{x^{14} y^{15} z^5}
\][/tex]
