Final answer:
The expression equivalent to (4x−3y4)−2 is 1 over 16 times x raised to the sixth power times y raised to the eighth power, or 1 / (16x6y8). This is achieved by applying the negative exponent rule to every part of the expression, including the coefficient.
Step-by-step explanation:
The student is asking about the equivalence of a mathematical expression with exponents. Specifically, they want to understand which expression is equivalent to (4x−3y4)−2, where the original expression involves a subtraction inside parentheses raised to a negative exponent.
Negative exponents indicate that the base of the power should be taken as the reciprocal, according to the identity x−n = 1 / xn. When applying a negative exponent to an entire expression inside parentheses, it indicates that each term inside the parentheses will be raised to that power, including the coefficient. Therefore, raising (4x3) to the power of −2 would result in 1 / (4x3)2, and similarly for the y term. Multiplication and division rules indicate that when two negative numbers multiply, the result is positive. Accordingly, when raising a negative number to an even power, the result is positive.
Thus, the answer is 1 over quantity 16 times x raised to the sixth power times y raised to the eighth power end quantity, which is expressed as 1 / (16x6y8).