In the following find the approximate values, using differentials :
`(26.57)^(1//3)`

Answer

To approximate the value of \( (26.57)^{1/3} \) using differentials, we can follow these steps:

### Step 1: Define the function
Let \( y = x^{1/3} \). We want to find \( y \) when \( x = 26.57 \).

### Step 2: Identify a nearby point
We know that \( 27^{1/3} = 3 \). So, we will use \( x = 27 \) as our nearby point.

### Step 3: Calculate \( \Delta x \)
Calculate \( \Delta x \) as follows:
\[ \Delta x = 26.57 - 27 = -0.43 \]

### Step 4: Differentiate the function
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{3} x^{-2/3} \]

### Step 5: Evaluate the derivative at \( x = 27 \)
Substituting \( x = 27 \) into the derivative:
\[ \frac{dy}{dx} \bigg|_{x=27} = \frac{1}{3} \cdot 27^{-2/3} \]
Since \( 27^{1/3} = 3 \), we have:
\[ 27^{-2/3} = \frac{1}{(27^{1/3})^2} = \frac{1}{3^2} = \frac{1}{9} \]
Thus,
\[ \frac{dy}{dx} \bigg|_{x=27} = \frac{1}{3} \cdot \frac{1}{9} = \frac{1}{27} \]

### Step 6: Calculate \( \Delta y \)
Using the formula \( \Delta y \approx \frac{dy}{dx} \Delta x \):
\[ \Delta y \approx \frac{1}{27} \cdot (-0.43) = -\frac{0.43}{27} \]

### Step 7: Find \( y \)
Now, we can find the approximate value of \( (26.57)^{1/3} \):
\[ y \approx 3 + \Delta y = 3 - \frac{0.43}{27} \]
Calculating \( \frac{0.43}{27} \):
\[ \frac{0.43}{27} \approx 0.0159 \]
Thus,
\[ y \approx 3 - 0.0159 \approx 2.9841 \]

### Conclusion
The approximate value of \( (26.57)^{1/3} \) is approximately \( 2.984 \).

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