Using differentials, find the approximate values of the following:
`(i) root(4)15 (ii)(82)^(1//4)`

Answer

To solve the problem using differentials, we will approximate the values of \(15^{1/4}\) and \(82^{1/4}\).

### Part (i): Approximate \(15^{1/4}\)

1. **Define the function**: Let \(y = x^{1/4}\). We want to find \(y\) when \(x = 15\).

2. **Choose a nearby value**: We can choose \(x = 16\) because it is close to 15 and easy to compute. Thus, we have:
- \(x = 16\)
- \(\Delta x = 15 - 16 = -1\)

3. **Calculate \(y\) at \(x = 16\)**:
\[ y = 16^{1/4} = (2^4)^{1/4} = 2 \]

4. **Find the derivative**: We differentiate \(y = x^{1/4}\):
\[ \frac{dy}{dx} = \frac{1}{4} x^{-3/4} \]

5. **Evaluate the derivative at \(x = 16\)**:
\[ \frac{dy}{dx} \bigg|_{x=16} = \frac{1}{4} \cdot 16^{-3/4} = \frac{1}{4} \cdot \left(\frac{1}{2^3}\right) = \frac{1}{4} \cdot \frac{1}{8} = \frac{1}{32} \]

6. **Calculate \(\Delta y\)**: Since \(\Delta x = -1\):
\[ \Delta y = \frac{dy}{dx} \cdot \Delta x = \frac{1}{32} \cdot (-1) = -\frac{1}{32} \]

7. **Approximate \(15^{1/4}\)**:
\[ 15^{1/4} \approx y + \Delta y = 2 - \frac{1}{32} = \frac{64}{32} - \frac{1}{32} = \frac{63}{32} \]

### Part (ii): Approximate \(82^{1/4}\)

1. **Define the function**: Let \(y = x^{1/4}\). We want to find \(y\) when \(x = 82\).

2. **Choose a nearby value**: We can choose \(x = 81\) because it is close to 82 and easy to compute. Thus, we have:
- \(x = 81\)
- \(\Delta x = 82 - 81 = 1\)

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

4. **Find the derivative**: We already have the derivative from the previous part:
\[ \frac{dy}{dx} = \frac{1}{4} x^{-3/4} \]

5. **Evaluate the derivative at \(x = 81\)**:
\[ \frac{dy}{dx} \bigg|_{x=81} = \frac{1}{4} \cdot 81^{-3/4} = \frac{1}{4} \cdot \left(\frac{1}{3^3}\right) = \frac{1}{4} \cdot \frac{1}{27} = \frac{1}{108} \]

6. **Calculate \(\Delta y\)**: Since \(\Delta x = 1\):
\[ \Delta y = \frac{dy}{dx} \cdot \Delta x = \frac{1}{108} \cdot 1 = \frac{1}{108} \]

7. **Approximate \(82^{1/4}\)**:
\[ 82^{1/4} \approx y + \Delta y = 3 + \frac{1}{108} = \frac{324}{108} + \frac{1}{108} = \frac{325}{108} \]

### Final Answers:
- \(15^{1/4} \approx \frac{63}{32}\)
- \(82^{1/4} \approx \frac{325}{108}\)