`196.1xx196.1xx196.1xx4.01xx4.01xx4.001xx4.999xx4.999=196.1^(3)xx4xx?`

Options

A. 400
B. 16
C. 32
D. 64

Explanation

To solve the equation \(196.1 \times 196.1 \times 196.1 \times 4.01 \times 4.01 \times 4.001 \times 4.999 \times 4.999 = 196.1^3 \times 4 \times ?\), we will follow these steps:

### Step 1: Rewrite the equation
The left-hand side can be rewritten as:
\[ 196.1^3 \times (4.01 \times 4.01 \times 4.001 \times 4.999 \times 4.999) \]
This allows us to see that we can factor out \(196.1^3\) from both sides.

### Step 2: Simplify the 4.01, 4.01, 4.001, and 4.999
- **For \(4.01\)**: We can round it to \(4\) since \(0.01\) is small.
- **For \(4.01\)** (again): It remains \(4\).
- **For \(4.001\)**: This can also be rounded to \(4\).
- **For \(4.999\)**: This can be rounded to \(5\) because it is closer to \(5\) than \(4\).

### Step 3: Count the factors of \(4\) and \(5\)
Now we can express the product:
\[ 4 \times 4 \times 4 \times 5 \times 5 \]
This can be rewritten as:
\[ 4^3 \times 5^2 \]

### Step 4: Substitute back into the equation
Now we can substitute back into the equation:
\[ 196.1^3 \times (4^3 \times 5^2) = 196.1^3 \times 4 \times ? \]

### Step 5: Cancel \(196.1^3\) from both sides
We can cancel \(196.1^3\) from both sides:
\[ 4^3 \times 5^2 = 4 \times ? \]

### Step 6: Solve for \(?\)
Now we can isolate \(?\):
\[ ? = \frac{4^3 \times 5^2}{4} \]
This simplifies to:
\[ ? = 4^2 \times 5^2 \]

### Step 7: Calculate \(4^2\) and \(5^2\)
Calculating the powers:
- \(4^2 = 16\)
- \(5^2 = 25\)

### Step 8: Multiply the results
Now we multiply:
\[ ? = 16 \times 25 \]
Calculating this gives:
\[ ? = 400 \]

### Final Answer
Thus, the missing number is:
\[ \boxed{400} \]