`3 (3)/(5) xx 3 (3)/(5) + 2 xx 3 (3)/(5) xx (2)/(5) + (2)/(5) xx (2)/(5)` = ?

Options

A. 15
B. 16
C. 17
D. 18

Answer

To solve the given expression step by step, we will follow the simplification process outlined in the video transcript.

### Step 1: Convert Mixed Numbers to Improper Fractions
The mixed number \(3 \frac{3}{5}\) can be converted to an improper fraction:
\[ 3 \frac{3}{5} = \frac{3 \times 5 + 3}{5} = \frac{15 + 3}{5} = \frac{18}{5} \]
So, we can rewrite the expression as:
\[ \frac{18}{5} \times \frac{18}{5} + 2 \times \frac{18}{5} \times \frac{2}{5} + \frac{2}{5} \times \frac{2}{5} \]

### Step 2: Define Variables
Let:
\[ A = \frac{18}{5}, \quad B = \frac{2}{5} \]
Now, we can rewrite the expression using \(A\) and \(B\):
\[ A \times A + 2 \times A \times B + B \times B \]

### Step 3: Apply the Algebraic Identity
The expression \(A \times A + 2 \times A \times B + B \times B\) can be recognized as the expansion of \((A + B)^2\):
\[ A^2 + 2AB + B^2 = (A + B)^2 \]

### Step 4: Calculate \(A + B\)
Now we need to calculate \(A + B\):
\[ A + B = \frac{18}{5} + \frac{2}{5} = \frac{18 + 2}{5} = \frac{20}{5} = 4 \]

### Step 5: Square the Result
Now, we can find \((A + B)^2\):
\[ (A + B)^2 = 4^2 = 16 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{16} \]