The value of `(127)^(1//3)` to four decimal places, is

Options

A. `5.0267`
B. `5.4267`
C. `5.5267`
D. `5.001`

Explanation

To find the value of \( (127)^{1/3} \) to four decimal places using approximation methods, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Function**:
Let \( y = f(x) = x^{1/3} \).

2. **Identify the Point of Approximation**:
We choose \( x = 125 \) because it is a perfect cube (since \( 5^3 = 125 \)). Thus, we can write:
\[ 127^{1/3} = (125 + 2)^{1/3} \]

3. **Use the Differential Approximation**:
We can express this as:
\[ y + dy = f(x + dx) \]
where \( dx = 2 \) (since \( 127 - 125 = 2 \)).

4. **Calculate \( y \)**:
From our function:
\[ y = f(125) = 125^{1/3} = 5 \]

5. **Find \( dy \)**:
To find \( dy \), we first need to calculate the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{3} x^{-2/3} \]
Now substituting \( x = 125 \):
\[ \frac{dy}{dx} = \frac{1}{3} (125)^{-2/3} \]
Since \( 125^{1/3} = 5 \), then \( 125^{-2/3} = \frac{1}{5^2} = \frac{1}{25} \):
\[ \frac{dy}{dx} = \frac{1}{3} \cdot \frac{1}{25} = \frac{1}{75} \]

6. **Calculate \( dy \)**:
Now, we can find \( dy \):
\[ dy = \frac{dy}{dx} \cdot dx = \frac{1}{75} \cdot 2 = \frac{2}{75} \]

7. **Approximate \( \frac{2}{75} \)**:
To make calculations easier, we can multiply and divide by 4:
\[ dy = \frac{2 \cdot 4}{75 \cdot 4} = \frac{8}{300} \]
Simplifying \( \frac{8}{300} \):
\[ dy = \frac{2}{75} \approx 0.0267 \quad (\text{approximately}) \]

8. **Final Calculation**:
Now we can find \( 127^{1/3} \):
\[ 127^{1/3} \approx y + dy = 5 + 0.0267 = 5.0267 \]

### Conclusion:
Thus, the value of \( (127)^{1/3} \) to four decimal places is approximately:
\[ \boxed{5.0267} \]