Find the approximate value of `(26)^(1/3)` .

Answer

To find the approximate value of \( (26)^{1/3} \), we can use the method of derivatives. Here’s a step-by-step solution:

### Step 1: Define the function
Let \( y = f(x) = x^{1/3} \). We want to approximate \( f(26) \).

### Step 2: Choose a point close to 26
We choose \( x = 27 \) because \( 27 \) is a perfect cube, and we know \( f(27) = 27^{1/3} = 3 \).

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

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

### Step 5: Calculate the change in \( x \)
We have \( x = 27 \) and we want to find \( f(26) \). The change in \( x \) is:
\[ \Delta x = 26 - 27 = -1 \]

### Step 6: Use the linear approximation formula
Using the linear approximation formula:
\[ f(26) \approx f(27) + f'(27) \cdot \Delta x \]
Substituting the values:
\[ f(26) \approx 3 + \left(\frac{1}{27}\right)(-1) = 3 - \frac{1}{27} \]

### Step 7: Simplify the expression
Now, we simplify:
\[ f(26) \approx 3 - \frac{1}{27} = \frac{81}{27} - \frac{1}{27} = \frac{80}{27} \]

### Step 8: Calculate the approximate value
Now, we can calculate:
\[ \frac{80}{27} \approx 2.96296 \approx 2.963 \]

Thus, the approximate value of \( (26)^{1/3} \) is:
\[ \boxed{2.963} \]