`(26.57)^(1//3)`

Answer

To evaluate \( (26.57)^{1/3} \) using differentials, we can follow these steps:

### Step-by-Step Solution

1. **Identify a nearby value**:
We can express \( 26.57 \) as \( 27 - 0.43 \). Here, we take \( x = 27 \) and \( \Delta x = -0.43 \).

2. **Define the function**:
Let \( f(x) = x^{1/3} \). We need to evaluate \( f(x + \Delta x) \), which is \( f(26.57) \).

3. **Use the differential approximation**:
The formula for the differential approximation is:
\[ f(x + \Delta x) \approx f(x) + \Delta x \cdot f'(x) \]

4. **Calculate \( f(x) \)**:
We first find \( f(27) \):
\[ f(27) = 27^{1/3} = 3 \]

5. **Find the derivative \( f'(x) \)**:
The derivative of \( f(x) = x^{1/3} \) is:
\[ f'(x) = \frac{1}{3} x^{-2/3} \]
Now, we evaluate \( f'(27) \):
\[ f'(27) = \frac{1}{3} \cdot 27^{-2/3} = \frac{1}{3} \cdot \frac{1}{9} = \frac{1}{27} \]

6. **Substitute into the approximation formula**:
Now we substitute \( f(27) \), \( \Delta x \), and \( f'(27) \) into the differential approximation:
\[ f(26.57) \approx f(27) + \Delta x \cdot f'(27) \]
\[ f(26.57) \approx 3 + (-0.43) \cdot \frac{1}{27} \]

7. **Calculate the product**:
Now, calculate \( -0.43 \cdot \frac{1}{27} \):
\[ -0.43 \cdot \frac{1}{27} \approx -0.0159259 \approx -0.016 \]

8. **Final calculation**:
Now, we can find \( f(26.57) \):
\[ f(26.57) \approx 3 - 0.016 = 2.984 \]

### Final Answer:
Thus, \( (26.57)^{1/3} \approx 2.984 \).