Final answer:
The value of (125/216)^-4/3 is found by taking the reciprocal of 125/216, raising it to the fourth power, and then taking the cube root of that result. The final value is expected to be greater than 1 since 216 is greater than 125.
Step-by-step explanation:
The question asks us to find the value of the expression (125/216)^-4/3. To solve this, we need to apply the rules of exponents. Since we have a negative exponent, we know that this represents the reciprocal of the base raised to the positive exponent. Next, because our exponent is a fraction, the numerator represents the power and the denominator represents the root.
So, we will first convert our base, which is the fraction 125/216, to its reciprocal because of the negative exponent, and then we'll raise it to the power of 4 which is our numerator and take the cube root, the denominator 3, of the result. Let's break down the steps:
- Calculate the reciprocal of 125/216, which is 216/125.
- Raise this to the power of 4 to get (216/125)^4.
- Take the cube root of the result from step 2, which is the same as raising the result to the power of 1/3.
Here's the calculation:
- The reciprocal of 125/216 is 216/125.
- Raising this to the 4th power, (216/125)^4.
- Finally, taking the cube root, or raising to the 1/3 power, gives us the final answer.
Without actual calculations here as we should not use a calculator, we would expect the final result to be a number greater than 1 because 216 is greater than 125, and we are raising it to a positive power after taking the reciprocal.