Question

The approximate value of \((0.007)^{\frac{1}{3}}\) is

• A. \(\frac{21}{120}\)
• B. \(\frac{23}{120}\)
• C. \(\frac{29}{120}\)
• D. \(\frac{31}{120}\)

Hint:

Write 0.007 as 0.008 - 0.001 and use a small increment approximation.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
For small \(h\), \(f(a - h) \approx f(a) - h f'(a)\). We choose a nearby perfect cube, \(0.008 = (0.2)^3\).

Step 2: Apply the approximation
Take \(f(x) = x^{1/3}\), \(a = 0.008\), \(h = 0.001\). Then \(f(a) = 0.2\) and \(f'(a) = \dfrac{1}{3a^{2/3}} = \dfrac{1}{3 \times 0.04} = \dfrac{25}{3}\).
\[ (0.007)^{1/3} \approx 0.2 - 0.001 \times \frac{25}{3} = 0.2 - \frac{1}{120} \]
\[ = \frac{24}{120} - \frac{1}{120} = \frac{23}{120} \]

Step 3: Check options
Option (A) 21/120 = 0.175 and (C) 29/120 = 0.2417 are too far from 0.2. The exact cube root is about 0.1913, and 23/120 = 0.1917 is the closest.

Final Answer:
The approximate value is \(\frac{23}{120}\), option (B). \[ \boxed{\frac{23}{120}} \]