Sure! Let's simplify the given mathematical expression step-by-step. The expression to simplify is:
[tex]\[
(-5a^9)(2a^5b^{-3})
\][/tex]
1. Separate the constants, and then the variables with the same base:
[tex]\[
(-5) \cdot (2) \cdot (a^9) \cdot (a^5) \cdot (b^{-3})
\][/tex]
2. Multiply the constants:
[tex]\[
-5 \times 2 = -10
\][/tex]
3. Combine the terms with the same base:
- For the [tex]\(a\)[/tex] terms: [tex]\(a^9 \cdot a^5\)[/tex]
- For the [tex]\(b\)[/tex] term: not applicable here since there is only [tex]\(b^{-3}\)[/tex]
Using the properties of exponents, specifically [tex]\(x^m \cdot x^n = x^{m+n}\)[/tex]:
[tex]\[
a^9 \cdot a^5 = a^{9+5} = a^{14}
\][/tex]
4. Keep the [tex]\(b\)[/tex] term as it is with the exponent unchanged:
[tex]\[
b^{-3}
\][/tex]
5. Combine all simplified terms:
[tex]\[
-10 \cdot a^{14} \cdot b^{-3}
\][/tex]
The final simplified expression is:
[tex]\[
-10a^{14}b^{-3}
\][/tex]
Since the problem asks for the answer with positive exponents only, we need to convert [tex]\(b^{-3}\)[/tex] to its positive exponent counterpart:
Using the property [tex]\(x^{-n} = \frac{1}{x^n}\)[/tex]:
[tex]\[
b^{-3} = \frac{1}{b^3}
\][/tex]
So the expression [tex]\(-10a^{14}b^{-3}\)[/tex] can be rewritten as:
[tex]\[
-10a^{14}\frac{1}{b^3} = \frac{-10a^{14}}{b^3}
\][/tex]
Thus, the simplified expression with positive exponents is:
[tex]\[
\boxed{\frac{-10a^{14}}{b^3}}
\][/tex]
