Find the product `ab(a^(2) + b^(2))` and evaluate it for a = 2 and b = 3.

Options

A. 26
B. 78
C. 86
D. 54

Answer

To solve the problem, we need to find the product \( ab(a^2 + b^2) \) and then evaluate it for \( a = 2 \) and \( b = 3 \).

### Step-by-Step Solution:

1. **Write down the expression**:
\[ ab(a^2 + b^2) \]

2. **Distribute \( ab \) into the expression \( (a^2 + b^2) \)**:
\[ ab(a^2 + b^2) = ab \cdot a^2 + ab \cdot b^2 \]

3. **Simplify each term**:
- For the first term:
\[ ab \cdot a^2 = a^{1+2}b = a^3b \]
- For the second term:
\[ ab \cdot b^2 = a \cdot b^{1+2} = ab^3 \]
Thus, we can rewrite the expression as:
\[ a^3b + ab^3 \]

4. **Substitute \( a = 2 \) and \( b = 3 \)**:
\[ (2^3 \cdot 3) + (2 \cdot 3^3) \]

5. **Calculate \( 2^3 \) and \( 3^3 \)**:
- \( 2^3 = 8 \)
- \( 3^3 = 27 \)

6. **Substitute these values back into the expression**:
\[ (8 \cdot 3) + (2 \cdot 27) \]

7. **Calculate each term**:
- \( 8 \cdot 3 = 24 \)
- \( 2 \cdot 27 = 54 \)

8. **Add the results together**:
\[ 24 + 54 = 78 \]

### Final Answer:
The evaluated product \( ab(a^2 + b^2) \) for \( a = 2 \) and \( b = 3 \) is \( 78 \).

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