What is the product?

[tex]\[ (3a^2 b^7)(5a^3 b^8) \][/tex]

A. \( 8a^5 b^{15} \)

B. \( 8a^6 b^{56} \)

C. \( 15a^5 b^{15} \)

D. [tex]\( 15a^5 b^{56} \)[/tex]



Answer :

To solve the given problem, we need to multiply the expressions \(3a^2b^7\) and \(5a^3b^8\). We will do this step-by-step, addressing the coefficients, the \(a\) terms, and the \(b\) terms separately.

### Step-by-Step Solution:

1. Multiply the coefficients:
- The coefficients in the expressions are \(3\) and \(5\).
- The product of the coefficients is:
[tex]\[ 3 \times 5 = 15 \][/tex]

2. Multiply the \(a\) terms:
- The exponents of \(a\) in the expressions are \(2\) and \(3\).
- When you multiply terms with the same base, you add their exponents:
[tex]\[ a^2 \times a^3 = a^{2+3} = a^5 \][/tex]

3. Multiply the \(b\) terms:
- The exponents of \(b\) in the expressions are \(7\) and \(8\).
- Similarly, when you multiply terms with the same base, you add their exponents:
[tex]\[ b^7 \times b^8 = b^{7+8} = b^{15} \][/tex]

### Putting it all together:
Combining the results from each step, the product of the expressions \(3a^2b^7\) and \(5a^3b^8\) is:
[tex]\[ 15 a^5 b^{15} \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{15 a^5 b^{15}} \][/tex]