Using differentials, find the approximate value of : `root5(33)`.

Answer

To find the approximate value of \( \sqrt[5]{33} \) using differentials, we can follow these steps:

### Step 1: Define the function
Let \( y = x^{1/5} \). We want to find \( y \) when \( x = 33 \).

### Step 2: Choose a point close to 33
We can choose a value of \( x \) that is close to 33 and for which we know the fifth root. A good choice is \( x = 32 \), since \( 32 = 2^5 \) and \( \sqrt[5]{32} = 2 \).

### Step 3: Calculate the differential
The differential \( dy \) can be calculated using the formula:
\[ dy = \frac{dy}{dx} \cdot dx \]
First, we need to find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{5} x^{-4/5} \]
Now, substituting \( x = 32 \):
\[ \frac{dy}{dx} \bigg|_{x=32} = \frac{1}{5} \cdot (32)^{-4/5} \]

### Step 4: Calculate \( (32)^{-4/5} \)
Since \( 32 = 2^5 \):
\[ (32)^{-4/5} = (2^5)^{-4/5} = 2^{-4} = \frac{1}{16} \]
Thus,
\[ \frac{dy}{dx} \bigg|_{x=32} = \frac{1}{5} \cdot \frac{1}{16} = \frac{1}{80} \]

### Step 5: Determine the change in \( x \)
The change in \( x \) from 32 to 33 is:
\[ dx = 33 - 32 = 1 \]

### Step 6: Calculate the change in \( y \)
Now we can calculate \( dy \):
\[ dy = \frac{1}{80} \cdot 1 = \frac{1}{80} \]

### Step 7: Calculate the approximate value of \( y \)
We know that when \( x = 32 \), \( y = \sqrt[5]{32} = 2 \). Therefore, the approximate value of \( y \) when \( x = 33 \) is:
\[ y \approx 2 + dy = 2 + \frac{1}{80} \]
Calculating this gives:
\[ y \approx 2 + 0.0125 = 2.0125 \]

### Conclusion
Thus, the approximate value of \( \sqrt[5]{33} \) is \( 2.0125 \).

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