Final answer:
The expression ((32x)/(x))⁻⁶⁄⁵ simplifies to 1/4 by cancelling the x variables, applying the negative exponent, and then simplifying the powers of 2.
Step-by-step explanation:
To simplify the expression ((32x)/(x))⁻⁶⁄⁵, we can start by simplifying the inside of the parentheses. Since the x in the numerator and the denominator are the same, they will cancel each other out, leaving us with 32. Now, we apply the outer negative exponent to 32. A negative exponent means we take the reciprocal of the base. Here, the base is 32. We follow the rule from Eq. A.9 that an expression with a negative exponent flips to the denominator, so 32⁻⁶⁄⁵ becomes 1/(32⁶⁄⁵).
Next, taking the fifth root of 32 raised to the sixth power, we get the fifth root of (32¶) which simplifies to 32 raised to the 6/5 power. Since 32 is 2⁵, we can then illustrate simplification as (2⁵)⁶⁄⁵, which further simplifies to 2¹ⁱ⁵, because when we raise a power to a power, we multiply the exponents. Finally, we can simplify 2¹ⁱ⁵ to 2², which is 4.
Thus, the given expression simplifies to 1/4.