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 \( y = f(x) = x^{1/4} \).

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

### Step 3: Calculate \( \Delta x \)
Since we are approximating \( 255 \) using \( 256 \):
\[ \Delta x = 255 - 256 = -1 \]

### Step 4: Find the derivative \( \frac{dy}{dx} \)
Using the power rule for differentiation:
\[ \frac{dy}{dx} = \frac{1}{4} x^{-3/4} \]

### Step 5: Evaluate the derivative at \( x = 256 \)
Substituting \( x = 256 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x=256} = \frac{1}{4} \cdot 256^{-3/4} \]
Since \( 256 = 4^4 \), we can rewrite it:
\[ 256^{-3/4} = (4^4)^{-3/4} = 4^{-3} = \frac{1}{64} \]
Thus,
\[ \frac{dy}{dx} \bigg|_{x=256} = \frac{1}{4} \cdot \frac{1}{64} = \frac{1}{256} \]

### Step 6: Calculate \( \Delta y \)
Now, we can find \( \Delta y \):
\[ \Delta y = \frac{dy}{dx} \cdot \Delta x = \frac{1}{256} \cdot (-1) = -\frac{1}{256} \]

### Step 7: Find \( y \) at \( x = 256 \)
Calculate \( f(256) \):
\[ f(256) = 256^{1/4} = 4 \]

### Step 8: Approximate \( f(255) \)
Now we can approximate \( f(255) \):
\[ f(255) \approx f(256) + \Delta y = 4 - \frac{1}{256} \]
Calculating \( 4 - \frac{1}{256} \):
\[ 4 - \frac{1}{256} = \frac{1024}{256} - \frac{1}{256} = \frac{1023}{256} \approx 3.9961 \]

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

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