Let's simplify the expression [tex]\(a^6 b^3 \cdot a^2 b^{-2}\)[/tex] step by step.
Step 1: Combine the powers of [tex]\(a\)[/tex].
- We have [tex]\(a^6\)[/tex] and [tex]\(a^2\)[/tex].
- When multiplying terms with the same base, we add their exponents: [tex]\[a^6 \cdot a^2 = a^{6+2} = a^8.\][/tex]
Step 2: Combine the powers of [tex]\(b\)[/tex].
- We have [tex]\(b^3\)[/tex] and [tex]\(b^{-2}\)[/tex].
- When multiplying terms with the same base, we add their exponents:
[tex]\[b^3 \cdot b^{-2} = b^{3 + (-2)} = b^{3-2} = b^1.\][/tex]
Step 3: Write down the simplified expression.
[tex]\[a^6 b^3 \cdot a^2 b^{-2} = a^8 b.\][/tex]
Thus, the simplified expression is [tex]\(\boxed{a^8 b}\)[/tex].
So, the correct answer is:
C. [tex]\(a^8 b\)[/tex]
