Find the approximate values of :
`root(3)(26.96)`

Answer

To find the approximate value of \( \sqrt[3]{26.96} \), we can use the concept of derivatives. Here’s a step-by-step solution:

### Step 1: Identify the function and the point of approximation
We want to approximate \( \sqrt[3]{26.96} \). We can express this as:
\[ \sqrt[3]{26.96} = \sqrt[3]{27 - 0.04} \]
Let \( x = 27 \) and \( h = -0.04 \). We will use the function:
\[ f(x) = x^{1/3} \]

### Step 2: Calculate \( f(27) \)
We first evaluate \( f(27) \):
\[ f(27) = 27^{1/3} = 3 \]

### Step 3: Find the derivative \( f'(x) \)
Next, we need to find the derivative of \( f(x) \):
\[ f'(x) = \frac{1}{3} x^{-2/3} \]

### Step 4: Calculate \( f'(27) \)
Now, we evaluate the derivative at \( x = 27 \):
\[ f'(27) = \frac{1}{3} \cdot 27^{-2/3} \]
Calculating \( 27^{-2/3} \):
\[ 27^{-2/3} = \frac{1}{(27^{1/3})^2} = \frac{1}{3^2} = \frac{1}{9} \]
Thus,
\[ f'(27) = \frac{1}{3} \cdot \frac{1}{9} = \frac{1}{27} \]

### Step 5: Use the linear approximation formula
Using the linear approximation formula:
\[ f(x + h) \approx f(x) + f'(x) \cdot h \]
Substituting the values we have:
\[ f(27 - 0.04) \approx f(27) + f'(27) \cdot (-0.04) \]
This gives:
\[ f(27 - 0.04) \approx 3 + \frac{1}{27} \cdot (-0.04) \]

### Step 6: Calculate the approximate value
Calculating the term:
\[ \frac{1}{27} \cdot (-0.04) = -\frac{0.04}{27} \]
Now, we need to compute \( -\frac{0.04}{27} \):
\[ -\frac{0.04}{27} \approx -0.00148148 \]
Thus, we have:
\[ f(27 - 0.04) \approx 3 - 0.00148148 \approx 2.99851852 \]

### Final Answer
Therefore, the approximate value of \( \sqrt[3]{26.96} \) is:
\[ \sqrt[3]{26.96} \approx 2.9985 \]

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