(x⁻³y⁻³)⁻³

Rationale

x^9y^9

To simplify the expression \((x^{-3}y^{-3})^{-3}\), we apply the power of a power property of exponents, which states that \((a^m)^n = a^{m \cdot n}\). Thus, we multiply the exponents, resulting in \(x^{9}y^{9}\).

A (1/(x⁹y⁹)) YOUR ANSWER

This choice represents the reciprocal of the correct answer. The original expression is not inverted; instead, it simplifies to a positive exponent form, which is \(x^9y^9\).

B (1/(x⁶y⁶)) YOUR ANSWER

Similar to choice A, this option suggests an incorrect reciprocal form that does not reflect the simplification of the original expression. The exponents are incorrect as they do not follow from the power multiplication.

C (x³y³) YOUR ANSWER

This choice incorrectly applies the power of a power rule. Instead of multiplying the exponents by \(-3\) and \(-3\) to get \(x^9y^9\), it erroneously results in lower exponents, which do not represent the original expression's transformation.

D (x⁶y⁶) YOUR ANSWER

This option also miscalculates the multiplication of the exponents. It suggests an incorrect application of the exponent rules and fails to account for the proper multiplication of \(-3\) by \(-3\), leading to an inadequate exponent result.

Conclusion

The simplification of the expression \((x^{-3}y^{-3})^{-3}\) correctly yields \(x^9y^9\). Each incorrect choice fails to accurately apply exponent rules, leading to misinterpretations of the original expression. The correct answer reflects the proper usage of exponent multiplication, confirming that the resulting expression retains positive exponents. This understanding is crucial for mastering algebraic manipulations involving exponents.

(x⁻³y⁻³)⁻³

x^9y^9

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