In the following find the approximate values, using differentials :
`(81.5)^(1//4)`

Answer

To find the approximate value of \( (81.5)^{1/4} \) using differentials, we can follow these steps:

### Step-by-Step Solution:

1. **Define the function**:
Let \( y = x^{1/4} \). We will use this function to find the approximate value of \( (81.5)^{1/4} \).

2. **Choose a point close to 81.5**:
We can choose \( x = 81 \) because it is close to \( 81.5 \).

3. **Calculate \( y \) at \( x = 81 \)**:
\[ y = 81^{1/4} = (3^4)^{1/4} = 3 \]

4. **Determine \( \Delta x \)**:
Since we are going from \( 81 \) to \( 81.5 \), we have:
\[ \Delta x = 81.5 - 81 = 0.5 \]

5. **Differentiate the function**:
We need to find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{4} x^{-3/4} \]

6. **Evaluate \( \frac{dy}{dx} \) at \( x = 81 \)**:
\[ \frac{dy}{dx} \bigg|_{x=81} = \frac{1}{4} (81)^{-3/4} \]
Since \( 81 = 3^4 \), we have:
\[ (81)^{-3/4} = (3^4)^{-3/4} = 3^{-3} = \frac{1}{27} \]
Thus,
\[ \frac{dy}{dx} \bigg|_{x=81} = \frac{1}{4} \cdot \frac{1}{27} = \frac{1}{108} \]

7. **Calculate \( \Delta y \)**:
Using the differential approximation:
\[ \Delta y \approx \frac{dy}{dx} \cdot \Delta x = \frac{1}{108} \cdot 0.5 = \frac{0.5}{108} = \frac{1}{216} \]

8. **Approximate \( y \) at \( x = 81.5 \)**:
Now we can find the approximate value of \( (81.5)^{1/4} \):
\[ (81.5)^{1/4} \approx y + \Delta y = 3 + \frac{1}{216} \]

9. **Calculate the final value**:
To get a numerical approximation:
\[ 3 + \frac{1}{216} \approx 3 + 0.00463 \approx 3.00463 \]

### Final Result:
Thus, the approximate value of \( (81.5)^{1/4} \) is approximately \( 3.00463 \).