In the following find the approximate values, using differentials :
`(15)^(1//4)`

Answer

To find the approximate value of \( 15^{1/4} \) using differentials, we can follow these steps:

### Step 1: Identify a nearby value
Let \( x = 16 \), since \( 16 \) is a perfect fourth power (i.e., \( 2^4 \)). We want to approximate \( f(x) = x^{1/4} \) at \( x = 15 \).

### Step 2: Calculate \( \delta x \)
Since \( 15 = x + \delta x \), we have:
\[ \delta x = 15 - 16 = -1 \]

### Step 3: Find the derivative \( \frac{dy}{dx} \)
We need to find the derivative of \( f(x) = x^{1/4} \):
\[ \frac{dy}{dx} = \frac{1}{4} x^{-3/4} = \frac{1}{4 \sqrt[4]{x^3}} \]

### Step 4: Evaluate the derivative at \( x = 16 \)
Now, we evaluate the derivative at \( x = 16 \):
\[ \frac{dy}{dx} \bigg|_{x=16} = \frac{1}{4 \cdot \sqrt[4]{16^3}} = \frac{1}{4 \cdot \sqrt[4]{4096}} = \frac{1}{4 \cdot 8} = \frac{1}{32} \]

### Step 5: Calculate \( \delta y \)
Using the formula \( \delta y = \frac{dy}{dx} \cdot \delta x \):
\[ \delta y = \frac{1}{32} \cdot (-1) = -\frac{1}{32} \]

### Step 6: Find the approximate value of \( 15^{1/4} \)
Now we can find the approximate value of \( 15^{1/4} \):
\[ 15^{1/4} \approx f(16) + \delta y \]
Since \( f(16) = 16^{1/4} = 2 \):
\[ 15^{1/4} \approx 2 - \frac{1}{32} = 2 - 0.03125 = 1.96875 \]

### Final Answer
Thus, the approximate value of \( 15^{1/4} \) is:
\[ \boxed{1.96875} \]