Final answer:
To simplify 64/125 raised to the power -2/3, we take the reciprocal of the base to get (125/64), take the cube root of each component, and then square the results to get 25/16.
Step-by-step explanation:
To simplify 64/125 raised to the power -2/3, we can break the problem into two main steps. First, remember that a negative exponent means we take the reciprocal of the base. Second, a fractional exponent indicates a root; specifically, the denominator of the fraction indicates which root to take (for example, 1/3 means cube root), and the numerator tells us the power the result should be raised to.
For 64/125 raised to the power of -2/3, taking the reciprocal gives us (125/64) and then taking the cube root and squaring that result follows: (125/64)^(1/3))^2.
The cube root of 125 is 5 and the cube root of 64 is 4. Then we square these numbers: (5^2)/(4^2), resulting in 25/16.