Use a linear approximation (or differentials) to estimate the given number. (Do not round your answer).(32.04)4/5

To estimate the given number using linear approximation or differentials, we first need to identify a function and a point close to the value we want to estimate.

The function that fits our problem is f(x) = x^(4/5). We want to estimate f(32.04), and a point close to 32.04 that we can easily compute is x = 32.

1. Compute the function at x = 32: f(32) = 32^(4/5) = 16.

2. Compute the derivative of the function: f'(x) = (4/5)x^(-1/5).

3. Compute the derivative at x = 32: f'(32) = (4/5)*32^(-1/5) = 0.4.

4. Use the linear approximation formula: Δf ≈ f'(a) * Δx, where Δx = x - a. In this case, Δx = 32.04 - 32 = 0.04.

5. Substitute the values into the formula: Δf ≈ 0.4 * 0.04 = 0.016.

6. Add this change to the original function value: f(32.04) ≈ f(32) + Δf = 16 + 0.016 = 16.016.

So, the estimate of (32.04)^(4/5) using linear approximation is 16.016.