In the following find the approximate values, using differentials :
`(28)^(1//3)`

Answer

To find the approximate value of \( (28)^{1/3} \) using differentials, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the function and point of approximation**:
Let \( y = f(x) = x^{1/3} \). We will approximate \( f(28) \) using a nearby value \( x = 27 \) (since \( 27 \) is a perfect cube).

2. **Calculate \( f(27) \)**:
\[ f(27) = 27^{1/3} = 3 \]

3. **Determine \( dx \)**:
Since we are moving from \( 27 \) to \( 28 \), we have:
\[ dx = 28 - 27 = 1 \]

4. **Find the derivative \( f'(x) \)**:
To find \( f'(x) \), we differentiate \( f(x) \):
\[ f'(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3 \sqrt[3]{x^2}} \]

5. **Evaluate \( f'(27) \)**:
Now, substituting \( x = 27 \) into the derivative:
\[ f'(27) = \frac{1}{3} \cdot 27^{-2/3} = \frac{1}{3} \cdot \frac{1}{9} = \frac{1}{27} \]

6. **Calculate \( dy \)**:
Using the formula \( dy = f'(x) \cdot dx \):
\[ dy = f'(27) \cdot dx = \frac{1}{27} \cdot 1 = \frac{1}{27} \]

7. **Approximate \( f(28) \)**:
Now we can approximate \( f(28) \):
\[ f(28) \approx f(27) + dy = 3 + \frac{1}{27} \]
To express this as a decimal:
\[ \frac{1}{27} \approx 0.037 \]
Thus,
\[ f(28) \approx 3 + 0.037 = 3.037 \]

### Final Answer:
The approximate value of \( (28)^{1/3} \) is \( \approx 3.037 \).