The value of the product `(7- (12)/(x)) ( 49 + (84)/(x) + (144)/(x ^(2))) at x =2 ` is

Options

A. 0
B. 559
C. 127
D. 128

Answer

To find the value of the product \((7 - \frac{12}{x})(49 + \frac{84}{x} + \frac{144}{x^2})\) at \(x = 2\), we will follow these steps:

### Step 1: Substitute \(x = 2\) into the expression
We start by substituting \(x = 2\) into the expression.

\[ (7 - \frac{12}{2})(49 + \frac{84}{2} + \frac{144}{2^2}) \]

### Step 2: Simplify the first part
Now, simplify the first part of the expression:

\[ 7 - \frac{12}{2} = 7 - 6 = 1 \]

### Step 3: Simplify the second part
Next, simplify the second part of the expression:

\[ 49 + \frac{84}{2} + \frac{144}{2^2} = 49 + 42 + \frac{144}{4} \]

Now, calculate \(\frac{144}{4}\):

\[ \frac{144}{4} = 36 \]

So, we have:

\[ 49 + 42 + 36 \]

### Step 4: Add the values together
Now, we will add these values together:

\[ 49 + 42 = 91 \]
\[ 91 + 36 = 127 \]

### Step 5: Calculate the final product
Now, we multiply the results from Step 2 and Step 4:

\[ 1 \times 127 = 127 \]

Thus, the value of the product at \(x = 2\) is:

\[ \boxed{127} \]

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