Final answer:
The expression (64/125)⁻²⁄³ simplifies to (125/64)²⁄³, which equals 25/16 after squaring and taking cube roots, with a final approximation of 1.56 to three significant figures.
Step-by-step explanation:
The question asks to evaluate (64/125)⁻²⁄³, which is a mathematical expression involving negative exponents and a fractional exponent. First, we can address the negative exponent by taking the reciprocal of the base. Thus, (64/125)⁻¹ = 125/64. We then take this result to the power of ²/³. In general, when dealing with exponents, the order of operations is important, as well as remembering that fractional exponents can be expressed as roots.
So, (64/125)⁻²⁄³ equals (125/64)²⁄³. Computing ²/³ as the cube root of something squared (or vice versa), we get ((125/64)¹¹)²⁄³ = √(125²) / √(64²)). So, we first square both the numerator and the denominator and then take the cube root. Squaring 125 gives 15625, and squaring 64 gives 4096. Taking the cube root results in approximately 25 / 16 after simplifying (since ³√15625 = 25 and ³√4096 = 16). The approximate value with three significant figures would be 1.56, which is the simplified and approximated answer to the original expression.