In the following find the approximate values, using differentials :
`(82)^(1//4)`

Answer

To find the approximate value of \( (82)^{1/4} \) using differentials, we can follow these steps:

### Step 1: Define the function
Let \( y = f(x) = x^{1/4} \).

### Step 2: Choose a point near 82
We can choose \( x = 81 \) because it is a perfect fourth power (since \( 81 = 3^4 \)). Thus, \( f(81) = (81)^{1/4} = 3 \).

### Step 3: Determine \( dx \)
We have \( dx = 82 - 81 = 1 \).

### Step 4: Calculate the derivative
We need to find the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{4} x^{-3/4} \]
Now, we will evaluate this derivative at \( x = 81 \):
\[ \frac{dy}{dx} \bigg|_{x=81} = \frac{1}{4} (81)^{-3/4} \]
Calculating \( (81)^{-3/4} \):
\[ (81)^{-3/4} = \frac{1}{(81)^{3/4}} = \frac{1}{(3^4)^{3/4}} = \frac{1}{3^3} = \frac{1}{27} \]
Thus,
\[ \frac{dy}{dx} \bigg|_{x=81} = \frac{1}{4} \cdot \frac{1}{27} = \frac{1}{108} \]

### Step 5: Calculate \( dy \)
Now, we can find \( dy \) using the formula:
\[ dy = \frac{dy}{dx} \cdot dx \]
Substituting the values we found:
\[ dy = \frac{1}{108} \cdot 1 = \frac{1}{108} \]

### Step 6: Approximate \( f(82) \)
Now, we can approximate \( f(82) \):
\[ f(82) \approx f(81) + dy = 3 + \frac{1}{108} \]
Calculating this gives:
\[ f(82) \approx 3 + 0.00926 \approx 3.00926 \]

### Final Answer
Thus, the approximate value of \( (82)^{1/4} \) is \( \approx 3.00926 \).

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