In the following find the approximate values, using differentials :
`(255)^(1//4)`

Answer

To find the approximate value of \( (255)^{1/4} \) using differentials, we can follow these steps:

### Step 1: Define the function
Let \( f(x) = x^{1/4} \). We want to find \( f(255) \).

### Step 2: Choose a point close to 255
Choose \( x = 256 \) because it is a perfect fourth power (i.e., \( 256 = 4^4 \)).

### Step 3: Calculate \( f(256) \)
Now, calculate \( f(256) \):
\[ f(256) = 256^{1/4} = 4 \]

### Step 4: Calculate the differential
Next, we need to find the derivative \( f'(x) \):
\[ f'(x) = \frac{1}{4} x^{-3/4} \]
Now, evaluate \( f'(256) \):
\[ f'(256) = \frac{1}{4} \cdot 256^{-3/4} \]
Since \( 256^{1/4} = 4 \), we have:
\[ 256^{-3/4} = \frac{1}{4^3} = \frac{1}{64} \]
Thus,
\[ f'(256) = \frac{1}{4} \cdot \frac{1}{64} = \frac{1}{256} \]

### Step 5: Calculate \( dx \)
Now, we set \( dx = 255 - 256 = -1 \).

### Step 6: Calculate \( dy \)
Using the formula \( dy = f'(x) \cdot dx \):
\[ dy = f'(256) \cdot (-1) = \frac{1}{256} \cdot (-1) = -\frac{1}{256} \]

### Step 7: Calculate \( f(255) \)
Now we can find \( f(255) \):
\[ f(255) \approx f(256) + dy = 4 - \frac{1}{256} \]
To calculate this, we convert \( 4 \) into a fraction:
\[ 4 = \frac{1024}{256} \]
Thus,
\[ f(255) \approx \frac{1024}{256} - \frac{1}{256} = \frac{1024 - 1}{256} = \frac{1023}{256} \]

### Step 8: Approximate the value
Now, we can calculate \( \frac{1023}{256} \):
\[ \frac{1023}{256} \approx 3.9961 \]

### Final Result
Therefore, the approximate value of \( (255)^{1/4} \) is:
\[ \boxed{3.9961} \]