Question 8

x⁹y⁹

To simplify the expression (x⁻³y⁻³)⁻³, we apply the rule of exponents which states that (a^m)^n = a^(m*n). Therefore, we multiply the exponents: -3 * -3, resulting in x⁹ and y⁹.

A) 1/(x⁹y⁹)

This option is incorrect because it represents the reciprocal of the correct answer. The simplification process does not lead to a negative exponent in this case, as we are simplifying the original expression rather than finding its reciprocal.

B) 1/(x⁶y⁶)

This option is also incorrect. Similar to option A, it suggests a negative exponent scenario that does not arise from the simplification of (x⁻³y⁻³)⁻³. The calculation yields positive exponents, not fractions.

C) x³y³

This option is incorrect because it reflects a misunderstanding of the exponent multiplication. The correct application of the exponent rule yields x⁹ and y⁹, not x³ and y³.

D) x⁶y⁶

This option is incorrect as well. It incorrectly applies the exponent rule by suggesting that the exponents should be halved instead of multiplied, which contradicts the proper simplification of the original expression.

E) x⁹y⁹

This is the correct answer, as it accurately reflects the result of the simplification process. By multiplying the exponents -3 and -3, we arrive at x⁹ and y⁹, confirming that this option is indeed correct.

Conclusion

The correct answer, x⁹y⁹, is the result of properly applying the laws of exponents to the original expression. All other options misinterpret the exponentiation process or propose incorrect results based on negative exponents, thereby reinforcing that option E is the only valid simplification.