The approximate value of `{3.92)^(2) + 3(2.1)^(4)}^(1//6)` is

Options

A. 2.040
B. 3.567
C. 1.562
D. 2.577

Answer

To find the approximate value of \((3.92^2 + 3 \cdot (2.1^4))^{1/6}\), we will use the concept of derivatives and linear approximation. Let's break it down step by step.

### Step 1: Calculate \(3.92^2\)

We can express \(3.92\) as \(3 + 0.92\):
\[ 3.92^2 = (3 + 0.92)^2 = 3^2 + 2 \cdot 3 \cdot 0.92 + (0.92)^2 \]
Calculating each term:
- \(3^2 = 9\)
- \(2 \cdot 3 \cdot 0.92 = 5.52\)
- \((0.92)^2 = 0.8464\)

Adding these together:
\[ 3.92^2 \approx 9 + 5.52 + 0.8464 = 15.3664 \]

### Step 2: Calculate \(2.1^4\)

We can express \(2.1\) as \(2 + 0.1\):
\[ 2.1^4 = (2 + 0.1)^4 \]
Using the binomial expansion:
\[ (2 + 0.1)^4 = 2^4 + 4 \cdot 2^3 \cdot 0.1 + 6 \cdot 2^2 \cdot (0.1)^2 + 4 \cdot 2 \cdot (0.1)^3 + (0.1)^4 \]
Calculating each term:
- \(2^4 = 16\)
- \(4 \cdot 2^3 \cdot 0.1 = 4 \cdot 8 \cdot 0.1 = 3.2\)
- \(6 \cdot 2^2 \cdot (0.1)^2 = 6 \cdot 4 \cdot 0.01 = 0.24\)
- \(4 \cdot 2 \cdot (0.1)^3 = 4 \cdot 2 \cdot 0.001 = 0.008\)
- \((0.1)^4 = 0.0001\)

Adding these together:
\[ 2.1^4 \approx 16 + 3.2 + 0.24 + 0.008 + 0.0001 = 19.4481 \]

### Step 3: Calculate \(3 \cdot (2.1^4)\)

Now, we multiply the result by 3:
\[ 3 \cdot 2.1^4 \approx 3 \cdot 19.4481 = 58.3443 \]

### Step 4: Combine the results

Now, we combine the two results:
\[ 3.92^2 + 3 \cdot (2.1^4) \approx 15.3664 + 58.3443 = 73.7107 \]

### Step 5: Take the sixth root

Finally, we need to take the sixth root of the combined result:
\[ (73.7107)^{1/6} \]

To approximate this, we can use the linear approximation:
Let \(f(x) = x^{1/6}\). We will evaluate it at \(x = 73\) and use the derivative:
\[ f'(x) = \frac{1}{6} x^{-5/6} \]
Calculating \(f(73)\):
\[ f(73) \approx 73^{1/6} \]
Using \(73^{1/6} \approx 2.04\) (as a rough estimate) and adjusting with the derivative:
\[ f(73.7107) \approx f(73) + f'(73) \cdot (73.7107 - 73) \]
Calculating \(f'(73)\):
\[ f'(73) = \frac{1}{6} \cdot 73^{-5/6} \approx \frac{1}{6} \cdot 0.054 \approx 0.009 \]
Thus:
\[ f(73.7107) \approx 2.04 + 0.009 \cdot 0.7107 \approx 2.04 + 0.0064 \approx 2.0464 \]

### Final Answer

The approximate value of \((3.92^2 + 3 \cdot (2.1^4))^{1/6}\) is approximately \(2.040\).

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