In the following find the approximate values, using differentials :
`root(3)(0.007)`

Answer

To find the approximate value of \( \sqrt[3]{0.007} \) using differentials, we can follow these steps:

### Step 1: Define the function
Let \( y = f(x) = x^{1/3} \). We want to approximate \( f(0.007) \).

### Step 2: Choose a point near 0.007
Select a point \( x = 0.008 \) which is close to \( 0.007 \).

### Step 3: Calculate \( \Delta x \)
Calculate \( \Delta x \):
\[ \Delta x = 0.007 - 0.008 = -0.001 \]

### Step 4: Find the derivative \( \frac{dy}{dx} \)
To find the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{3} x^{-2/3} \]

### Step 5: Evaluate the derivative at \( x = 0.008 \)
Now, evaluate \( \frac{dy}{dx} \) at \( x = 0.008 \):
\[ \frac{dy}{dx} \bigg|_{x=0.008} = \frac{1}{3} (0.008)^{-2/3} \]
Calculating \( (0.008)^{-2/3} \):
\[ 0.008 = 2^{-6} \implies (0.008)^{-2/3} = (2^{-6})^{-2/3} = 2^{4} = 16 \]
Thus,
\[ \frac{dy}{dx} \bigg|_{x=0.008} = \frac{1}{3} \times 16 = \frac{16}{3} \]

### Step 6: Calculate \( \Delta y \)
Now, calculate \( \Delta y \):
\[ \Delta y = \frac{dy}{dx} \cdot \Delta x = \frac{16}{3} \cdot (-0.001) = -\frac{16}{3000} = -\frac{8}{1500} = -\frac{4}{750} \approx -0.00533 \]

### Step 7: Calculate the approximate value of \( f(0.007) \)
Now, we can find the approximate value of \( f(0.007) \):
\[ f(0.007) \approx f(0.008) + \Delta y \]
First, calculate \( f(0.008) \):
\[ f(0.008) = (0.008)^{1/3} = 0.2 \]
Now, substitute:
\[ f(0.007) \approx 0.2 - 0.00533 \approx 0.19467 \]

### Final Answer
Thus, the approximate value of \( \sqrt[3]{0.007} \) is:
\[ \boxed{0.19467} \]