((x^3×y^-2)^-3 × (x^-1×y^2)^2) : (x^3×y4)^-2

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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Simplify the algebraic expression ((x^3×y^-2)^-3 × (x^-1×y^2)^2) : (x^3×y^4)^-2 step-by-step. Learn key exponentiation rules and how to handle negative exponents. Suitable for Grades 9-12 students.

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