To simplify and evaluate the given expression [tex]\(\left(11 j^{-3} k^{-2}\right)\left(j^3 k^4\right)\)[/tex], we will use the properties of exponents.
### Step-by-Step Solution:
1. Combine the expressions:
[tex]\[
\left(11 j^{-3} k^{-2}\right)\left(j^3 k^4\right)
\][/tex]
2. Apply the properties of exponents:
For any bases [tex]\(a\)[/tex] and any exponents [tex]\(m\)[/tex] and [tex]\(n\)[/tex], the property [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex] holds.
Using this property:
[tex]\[
j^{-3} \cdot j^3 = j^{(-3+3)} = j^0 = 1
\][/tex]
[tex]\[
k^{-2} \cdot k^4 = k^{(-2+4)} = k^2
\][/tex]
3. Simplify the expression:
[tex]\[
11 \cdot j^0 \cdot k^2 = 11 \cdot 1 \cdot k^2 = 11 k^2
\][/tex]
4. Substitute the given values:
- [tex]\(j = -8\)[/tex] (Note that [tex]\(j\)[/tex] is not needed as [tex]\(j^0 = 1\)[/tex])
- [tex]\(k = 7\)[/tex]
Now substitute [tex]\(k = 7\)[/tex] into the simplified expression:
[tex]\[
11 k^2 = 11 \cdot 7^2
\][/tex]
5. Evaluate [tex]\(k^2\)[/tex]:
[tex]\[
7^2 = 49
\][/tex]
6. Multiply the evaluated value by 11:
[tex]\[
11 \cdot 49 = 539
\][/tex]
### Final Answer:
When [tex]\(j = -8\)[/tex] and [tex]\(k = 7\)[/tex], the value of the expression is [tex]\(\boxed{539}\)[/tex].
