What is the value of `(4.6 + 3.1)^(2) - (4.6 - 3.1)^(2)` ?

Options

A. 54.68
B. 58.86
C. 53.32
D. 57.04

Answer

To solve the expression \((4.6 + 3.1)^{2} - (4.6 - 3.1)^{2}\), we can use the difference of squares formula. The difference of squares states that:

\[ a^2 - b^2 = (a + b)(a - b) \]

In our case, we can let:
- \(a = 4.6 + 3.1\)
- \(b = 4.6 - 3.1\)

### Step 1: Identify \(a\) and \(b\)

Calculate \(a\) and \(b\):

\[ a = 4.6 + 3.1 = 7.7 \]
\[ b = 4.6 - 3.1 = 1.5 \]

### Step 2: Apply the difference of squares formula

Now we apply the difference of squares formula:

\[ (4.6 + 3.1)^{2} - (4.6 - 3.1)^{2} = (a + b)(a - b) \]

### Step 3: Calculate \(a + b\) and \(a - b\)

Calculate \(a + b\) and \(a - b\):

\[ a + b = (4.6 + 3.1) + (4.6 - 3.1) = 7.7 + 1.5 = 9.2 \]
\[ a - b = (4.6 + 3.1) - (4.6 - 3.1) = 7.7 - 1.5 = 6.2 \]

### Step 4: Multiply the results

Now we multiply \(a + b\) and \(a - b\):

\[ (4.6 + 3.1)^{2} - (4.6 - 3.1)^{2} = 9.2 \times 6.2 \]

### Step 5: Calculate the final result

Now we perform the multiplication:

\[ 9.2 \times 6.2 = 57.04 \]

Thus, the value of \((4.6 + 3.1)^{2} - (4.6 - 3.1)^{2}\) is \(\boxed{57.04}\).