Final answer:
Applying the power rule, the expression ((-9x³z³)/(y⁴))³ simplifies to -729x¹z¹/y⁻¹² by cubing each part of the fraction and applying the power to each variable and coefficient.
Step-by-step explanation:
Applying the Power Rule to an Exponential Expression:
To apply the power rule to the expression ((-9x³z³)/(y⁴))³, you must raise each part of the fraction to the third power. Start by cubing the coefficient -9 to obtain -729. Next, apply the power of 3 to each variable separately, multiplying the exponents: x³ raised to the third power is x¹, and z³ raised to the third power is z¹. Since y⁴ is in the denominator, its exponent is negative; raising y⁴ to the third power gives us y⁻¹². As a result, the simplified expression is -729x¹z¹/y⁻¹².
Remember that a negative exponent indicates division. Negative exponents, such as x⁻⁴, equate to 1 divided by x´, flipping the base to the denominator; hence, y⁻¹² becomes 1/y⁻¹².