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To approximate the fourth root of 255, \( \sqrt[4]{255} \), using differentials, we can apply the following method:

### Step 1: Define the Function
Define \( f(x) = \sqrt[4]{x} = x^{1/4} \).

We want to approximate \( f(255) \), which is close to \( f(256) \), a number for which we know the exact fourth root:
\[
f(256) = \sqrt[4]{256} = 4
\]

### Step 2: Use Differentials
The differential \( df \) can be used to approximate the change in \( f(x) \) when moving from \( x = 256 \) to \( x = 255 \).

1. **Calculate the derivative**:
   \[
   f'(x) = \frac{1}{4}x^{-\frac{3}{4}} = \frac{1}{4x^{3/4}}
   \]
   
   At \( x = 256 \):
   \[
   f'(256) = \frac{1}{4 \times 256^{3/4}} = \frac{1}{4 \times 64} = \frac{1}{256}
   \]

2. **Calculate \( dx \)**:
   Since \( x \) changes from 256 to 255, \( dx = 255 - 256 = -1 \).

3. **Calculate \( df \)**:
   \[
   df = f'(256) \cdot dx = \frac{1}{256} \times (-1) = -\frac{1}{256}
   \]

4. **Approximate \( f(255) \)**:
   \[
   f(255) \approx f(256) + df = 4 - \frac{1}{256} = 4 - 0.00390625 = 3.99609375
   \]

### Calculator Comparison
Using a calculator, \( \sqrt[4]{255} \approx 3.996115 \) (rounded to six decimal places).

### Summary of Results
- **Using Differentials**: 3.996094 (rounded to six decimal places)
- **Using Calculator**: 3.996115

Would you like further details on the differential method or more practice problems?

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**Related Questions:**
1. How would we use differentials to approximate square roots?
2. What other approximation methods are commonly used besides differentials?
3. Can differentials be used for approximating trigonometric functions?
4. How accurate are differentials compared to Taylor series expansions?
5. Why does the differential approximation work best for values close to a known point?

**Tip:** When using differentials, the approximation improves if the point is very close to the known value, as larger distances increase the error.

Use differentials to approximate the value of the expression 4th root of 255 and compare the result with a calculator's answer, rounding to six decimal places.

Learn how to approximate the fourth root of 255 using differentials and compare the result with a calculator's answer. This step-by-step solution involves calculus concepts like differential approximations and derivatives. Ideal for college calculus students.

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