We can simplify the given expression step by step. The expression is:

$\frac{\left((x^3 \times y^{-2})^{-3} \times (x^{-1} \times y^2)^2\right)}{(x^3 \times y^4)^{-2}}$

Step 1: Apply exponents inside the parentheses.

We'll first apply the exponents inside the parentheses for each term.

1. $\left(x^3 \times y^{-2}\right)^{-3}$ means raising both $x^3$ and $y^{-2}$ to the power of $-3$: $(x^3)^{-3} = x^{-9}, \quad (y^{-2})^{-3} = y^6$ So, $(x^3 \times y^{-2})^{-3} = x^{-9} \times y^6$

2. $\left(x^{-1} \times y^2\right)^2$ means raising both $x^{-1}$ and $y^2$ to the power of $2$: $(x^{-1})^2 = x^{-2}, \quad (y^2)^2 = y^4$ So, $(x^{-1} \times y^2)^2 = x^{-2} \times y^4$

3. $\left(x^3 \times y^4\right)^{-2}$ means raising both $x^3$ and $y^4$ to the power of $-2$: $(x^3)^{-2} = x^{-6}, \quad (y^4)^{-2} = y^{-8}$ So, $(x^3 \times y^4)^{-2} = x^{-6} \times y^{-8}$

Step 2: Rewrite the expression using the simplified terms.

Now, substitute the simplified terms into the original expression:

$\frac{(x^{-9} \times y^6) \times (x^{-2} \times y^4)}{x^{-6} \times y^{-8}}$

Step 3: Combine the terms in the numerator.

Use the property of exponents, $a^m \times a^n = a^{m+n}$, to combine like terms in the numerator:

$(x^{-9} \times x^{-2}) = x^{-11}, \quad (y^6 \times y^4) = y^{10}$ So the numerator becomes: $x^{-11} \times y^{10}$

Step 4: Simplify the entire expression.

Now simplify the entire expression by dividing the numerator by the denominator. Use the property $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{x^{-11}}{x^{-6}} = x^{-11 - (-6)} = x^{-11 + 6} = x^{-5}$ $\frac{y^{10}}{y^{-8}} = y^{10 - (-8)} = y^{10 + 8} = y^{18}$

Final Answer:

The simplified expression is: $x^{-5} \times y^{18} = \frac{y^{18}}{x^5}$

Let me know if you need any clarifications or further details!

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