In the following find the approximate values, using differentials :
`((17)/(81))^(1//4)`.

Answer

To find the approximate value of \(\left(\frac{17}{81}\right)^{\frac{1}{4}}\) using differentials, we can follow these steps:

### Step 1: Define the function
Let \( y = f(x) = x^{\frac{1}{4}} \). We will approximate \( f\left(\frac{17}{81}\right) \) using the value of \( f\) at a nearby point.

### Step 2: Choose a nearby point
We can choose \( x = \frac{16}{81} \) because it is close to \( \frac{17}{81} \) and is easier to compute.

### Step 3: Calculate \( f\left(\frac{16}{81}\right) \)
Now, we calculate:
\[ f\left(\frac{16}{81}\right) = \left(\frac{16}{81}\right)^{\frac{1}{4}} = \frac{2}{3} \]
This is because \( 16 = 2^4 \) and \( 81 = 3^4 \), so:
\[ \left(\frac{16}{81}\right)^{\frac{1}{4}} = \frac{2^{4/4}}{3^{4/4}} = \frac{2}{3} \]

### Step 4: Calculate the differential
Next, we need to find \( dy \). First, we compute the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{4} x^{-\frac{3}{4}} \]
Now, substituting \( x = \frac{16}{81} \):
\[ \frac{dy}{dx} = \frac{1}{4} \left(\frac{16}{81}\right)^{-\frac{3}{4}} = \frac{1}{4} \cdot \left(\frac{2}{3}\right)^{-3} = \frac{1}{4} \cdot \frac{27}{8} = \frac{27}{32} \]

### Step 5: Calculate \( dx \)
Now, we find \( dx \):
\[ dx = \frac{17}{81} - \frac{16}{81} = \frac{1}{81} \]

### Step 6: Calculate \( dy \)
Now, we can find \( dy \):
\[ dy = \frac{dy}{dx} \cdot dx = \frac{27}{32} \cdot \frac{1}{81} = \frac{27}{2592} = \frac{1}{96} \]

### Step 7: Approximate the value
Now we can approximate \( f\left(\frac{17}{81}\right) \):
\[ f\left(\frac{17}{81}\right) \approx f\left(\frac{16}{81}\right) + dy = \frac{2}{3} + \frac{1}{96} \]

### Step 8: Find a common denominator
To add these fractions, we find a common denominator:
\[ \frac{2}{3} = \frac{64}{96} \]
Thus,
\[ f\left(\frac{17}{81}\right) \approx \frac{64}{96} + \frac{1}{96} = \frac{65}{96} \]

### Final Answer
The approximate value of \(\left(\frac{17}{81}\right)^{\frac{1}{4}}\) is \(\frac{65}{96}\).

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