To determine the value of the given expression [tex]\(\left(6 m^{-1} n^0\right)^{-3}\)[/tex] when [tex]\(m = 3\)[/tex] and [tex]\(n = -5\)[/tex], follow these steps:
1. Evaluate [tex]\(n^0\)[/tex]:
[tex]\[
n^0 = (-5)^0 = 1 \quad \text{(because any number to the power of 0 is 1)}
\][/tex]
2. Evaluate [tex]\(m^{-1}\)[/tex]:
[tex]\[
m^{-1} = 3^{-1} = \frac{1}{3}
\][/tex]
3. Substitute these values back into the expression:
[tex]\[
6 \cdot m^{-1} \cdot n^0 = 6 \cdot \frac{1}{3} \cdot 1 = 6 \cdot \frac{1}{3} = 2
\][/tex]
4. Now raise the result to the power of [tex]\(-3\)[/tex]:
[tex]\[
(2)^{-3} = \left(\frac{1}{2}\right)^3 = \frac{1}{2^3} = \frac{1}{8}
\][/tex]
Therefore, the value of the expression when [tex]\(m = 3\)[/tex] and [tex]\(n = -5\)[/tex] is:
[tex]\[
\frac{1}{8}
\][/tex]
Among the given choices:
- [tex]\(-8\)[/tex]
- [tex]\(-\frac{1}{8}\)[/tex]
- [tex]\(\frac{1}{8}\)[/tex]
- 8
The correct answer is:
[tex]\[
\boxed{\frac{1}{8}}
\][/tex]
