Answer :

Let's solve the expression step-by-step:

Given expression:
[tex]\[
\left(x^{-4} y^3 \cdot x^7 y^{-9}\right)^{-2}
\][/tex]

Step 1: Simplify the expression inside the parentheses

Start by combining the terms with the same base:

- For [tex]\(x^{-4} \cdot x^7\)[/tex], use the rule [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
x^{-4} \cdot x^7 = x^{-4+7} = x^3
\][/tex]

- For [tex]\(y^3 \cdot y^{-9}\)[/tex], similar rule applies:
[tex]\[
y^3 \cdot y^{-9} = y^{3-9} = y^{-6}
\][/tex]

Now, the expression inside the parentheses becomes:
[tex]\[
(x^3 y^{-6})
\][/tex]

Step 2: Apply the negative exponent outside the parentheses

The expression [tex]\((x^3 y^{-6})^{-2}\)[/tex] means that each component inside should be raised to the power of [tex]\(-2\)[/tex]:

- For [tex]\(x^3\)[/tex], use the rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(x^3)^{-2} = x^{3 \cdot (-2)} = x^{-6}
\][/tex]

- For [tex]\(y^{-6}\)[/tex]:
[tex]\[
(y^{-6})^{-2} = y^{-6 \cdot (-2)} = y^{12}
\][/tex]

Step 3: Combine the results

The expression simplifies to:
[tex]\[
x^{-6} \cdot y^{12}
\][/tex]

Final Answer:
The simplified expression is:
[tex]\[
x^{-6} y^{12}
\][/tex]

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Rewritten by : Barada