Sure, let's simplify the given expression step by step.
Given expression:
[tex]\[
\frac{\left(2 x^{-1} y^{-3}\right)^3}{\left(x^{-2} y^{-5}\right)^{-2}}
\][/tex]
### Step 1: Simplify the numerator
First, let's simplify the numerator \(\left(2 x^{-1} y^{-3}\right)^3\):
[tex]\[
(2 x^{-1} y^{-3})^3
= 2^3 \cdot (x^{-1})^3 \cdot (y^{-3})^3
= 8 x^{-3} y^{-9}
\][/tex]
So the numerator simplifies to:
[tex]\[
8 x^{-3} y^{-9}
\][/tex]
### Step 2: Simplify the denominator
Now, let's simplify the denominator \(\left(x^{-2} y^{-5}\right)^{-2}\):
[tex]\[
(x^{-2} y^{-5})^{-2}
= (x^{-2})^{-2} \cdot (y^{-5})^{-2}
= x^{4} \cdot y^{10}
\][/tex]
So the denominator simplifies to:
[tex]\[
x^4 y^{10}
\][/tex]
### Step 3: Simplify the whole fraction
Now, we put the simplified numerator and denominator together and simplify the fraction:
[tex]\[
\frac{8 x^{-3} y^{-9}}{x^4 y^{10}}
\][/tex]
This can be simplified by subtracting the exponents of the same bases:
[tex]\[
8 \cdot x^{-3 - 4} \cdot y^{-9 - 10}
= 8 \cdot x^{-7} \cdot y^{-19}
\][/tex]
### Step 4: Convert to positive exponents
Finally, we rewrite the expression using only positive exponents:
[tex]\[
8 \cdot x^{-7} \cdot y^{-19}
=
\frac{8}{x^{7} y^{19}}
\][/tex]
So, the simplified expression using only positive exponents is:
[tex]\[
\frac{8}{x^{7} y^{19}}
\][/tex]
