Question

Find the approximate value of \(\sqrt[3]{63}\).

Hint:

To divide by 48 quickly: \(1/48\) is slightly more than \(1/50 = 0.02\). Since \(48<50\), the value is slightly larger than \(0.02\), approximately \(0.021\). This estimation helps in rejecting distant options.

Answer 1

Step 1: Understanding the Question:
We need to find the approximate value of the cube root of 63. We use the differential method, choosing a nearby perfect cube (\( 4^3 = 64 \)).
Step 2: Key Formula or Approach:
Differential approximation: \( f(x + \Delta x) \approx f(x) + f'(x) \Delta x \).
Step 3: Detailed Explanation:
Let \( f(x) = x^{1/3} \).
Choose \( x = 64 \) and \( \Delta x = -1 \).
\( f(64) = 64^{1/3} = 4 \).
Derivative \( f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3(x^{1/3})^2} \).
At \( x = 64 \):
\[ f'(64) = \frac{1}{3(4^2)} = \frac{1}{48} \approx 0.020833 \]
Applying the approximation:
\[ f(63) \approx f(64) + f'(64)(-1) \]
\[ f(63) \approx 4 - 0.020833 = 3.979167 \]
Rounding to 4 decimal places gives \( 3.9792 \).
Step 4: Final Answer:
The approximate value is \( 3.9792 \).