To determine the value of the given expression when [tex]\( j = -2 \)[/tex] and [tex]\( k = -1 \)[/tex], we'll evaluate it step by step. The expression is:
[tex]\[
\left(\frac{j k^{-2}}{j^{-1} k^{-3}}\right)^3
\][/tex]
First, let's compute the inner fraction.
1. Calculate [tex]\( j k^{-2} \)[/tex]:
[tex]\[
j = -2, \quad k^{-2} = (-1)^{-2} = 1
\][/tex]
[tex]\[
j k^{-2} = -2 \times 1 = -2
\][/tex]
2. Calculate [tex]\( j^{-1} \)[/tex] and [tex]\( k^{-3} \)[/tex]:
[tex]\[
j^{-1} = (-2)^{-1} = -\frac{1}{2}
\][/tex]
[tex]\[
k^{-3} = (-1)^{-3} = -1
\][/tex]
[tex]\[
j^{-1} k^{-3} = -\frac{1}{2} \times -1 = \frac{1}{2}
\][/tex]
3. Calculate the inner fraction [tex]\( \frac{j k^{-2}}{j^{-1} k^{-3}} \)[/tex]:
[tex]\[
\frac{j k^{-2}}{j^{-1} k^{-3}} = \frac{-2}{\frac{1}{2}} = -2 \times 2 = -4
\][/tex]
4. Raise the result to the power of 3:
[tex]\[
(-4)^3 = -4 \times -4 \times -4 = -64
\][/tex]
Thus, the value of the expression [tex]\(\left(\frac{j k^{-2}}{j^{-1} k^{-3}}\right)^3\)[/tex] when [tex]\( j = -2 \)[/tex] and [tex]\( k = -1 \)[/tex] is:
[tex]\[
\boxed{-64}
\][/tex]
