The value of ` ((3.2)^3 - 0.008)/((3.2)^2 + 0.64 + 0.04)` is

Options

A. 0
B. 2.994
C. 3.208
D. 3

Answer

To solve the expression \(\frac{(3.2)^3 - 0.008}{(3.2)^2 + 0.64 + 0.04}\), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{(3.2)^3 - 0.008}{(3.2)^2 + 0.64 + 0.04} \]

### Step 2: Recognize \(0.008\) as \((0.2)^3\)
We can rewrite \(0.008\) as:
\[ 0.008 = (0.2)^3 \]
Thus, we can express the numerator as:
\[ (3.2)^3 - (0.2)^3 \]

### Step 3: Recognize the denominator
Next, we express the denominator:
\[ (3.2)^2 + 0.64 + 0.04 \]
Notice that \(0.64 = (0.8)^2\) and \(0.04 = (0.2)^2\). Hence, we can rewrite the denominator as:
\[ (3.2)^2 + (0.8)^2 + (0.2)^2 \]

### Step 4: Use the difference of cubes formula
The numerator can be factored using the difference of cubes formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Let \(a = 3.2\) and \(b = 0.2\). Therefore, we have:
\[ (3.2)^3 - (0.2)^3 = (3.2 - 0.2)((3.2)^2 + (3.2)(0.2) + (0.2)^2) \]

### Step 5: Substitute back into the expression
Now substituting back into the expression gives:
\[ \frac{(3.2 - 0.2)((3.2)^2 + (3.2)(0.2) + (0.2)^2)}{(3.2)^2 + (0.8)^2 + (0.2)^2} \]

### Step 6: Simplify the expression
Notice that the term \((3.2)^2 + (3.2)(0.2) + (0.2)^2\) in the numerator is the same as the denominator:
\[ (3.2)^2 + (0.8)^2 + (0.2)^2 \]
Thus, they cancel out, leaving us with:
\[ 3.2 - 0.2 \]

### Step 7: Calculate the final value
Now we can calculate:
\[ 3.2 - 0.2 = 3.0 \]

### Final Answer
The value of the expression is:
\[ \boxed{3} \]