To simplify the expression
[tex]\[
\left(\frac{2 y^{-2} z^{-3}}{x^{-3} y^{-2} z^{-1}}\right)^{-5},
\][/tex]
we will proceed step-by-step.
### Step 1: Simplify the inner fraction
First, simplify the fraction inside the parentheses:
[tex]\[
\frac{2 y^{-2} z^{-3}}{x^{-3} y^{-2} z^{-1}}.
\][/tex]
### Step 2: Combine the exponents
We combine the like terms by canceling out the common terms in the numerator and the denominator.
Since [tex]\(y^{-2}\)[/tex] appears in both the numerator and the denominator, they cancel each other out:
[tex]\[
\frac{y^{-2}}{y^{-2}} = 1.
\][/tex]
For the remaining terms, we use the laws of exponents:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}.
\][/tex]
Applying this to the [tex]\(z\)[/tex] terms:
[tex]\[
\frac{z^{-3}}{z^{-1}} = z^{-3 - (-1)} = z^{-3 + 1} = z^{-2}.
\][/tex]
For the [tex]\(x\)[/tex] terms:
[tex]\[
\frac{1}{x^{-3}} = x^3.
\][/tex]
### Step 3: Simplify the fraction entirely
After simplifying, we have:
[tex]\[
\frac{2 z^{-3}}{x^{-3} z^{-1}} = 2 \cdot x^3 \cdot z^{-2}.
\][/tex]
### Step 4: Apply the negative exponent
Now we apply the negative exponent of [tex]\(-5\)[/tex] to the simplified expression:
[tex]\[
\left(2 x^3 z^{-2}\right)^{-5}.
\][/tex]
Applying the exponent rule [tex]\((ab)^n = a^n b^n\)[/tex]:
[tex]\[
2^{-5} \cdot (x^3)^{-5} \cdot (z^{-2})^{-5}.
\][/tex]
We simplify each term individually:
[tex]\[
2^{-5} = \frac{1}{2^5} = \frac{1}{32},
\][/tex]
[tex]\[
(x^3)^{-5} = x^{3 \cdot (-5)} = x^{-15},
\][/tex]
[tex]\[
(z^{-2})^{-5} = z^{-2 \cdot (-5)} = z^{10}.
\][/tex]
### Step 5: Combine the results
Combining these results, we get:
[tex]\[
\frac{1}{32} x^{-15} z^{10}.
\][/tex]
To express this as a single fraction:
[tex]\[
\frac{z^{10}}{32 x^{15}}.
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{\frac{z^{10}}{32 x^{15}}}.
\][/tex]
