Comparing Mathematical Expressions: Finding Greater Value When b > 1

To solve this problem, let's simplify and compare the given expressions one by one.

• Simplification of each expression:
• $b^3 \times b^5 \times b^{-2} = b^{3+5-2} = b^6$
• $b^7 \times b^4 = b^{7+4} = b^{11}$
• $(b)^3 \times b^4 = b^{3+4} = b^7$
• $b^{-3} \times b^6 = b^{-3+6} = b^3$

Next, we compare the simplified exponents:
- The first expression simplifies to $b^6$.
- The second expression simplifies to $b^{11}$.
- The third expression simplifies to $b^7$.
- The fourth expression simplifies to $b^3$.

Among these, $b^{11}$ is the greatest because exponent 11 is the highest. Since $b > 1$, greater exponents correspond to greater values.

Therefore, the expression with the greatest value is $b^7 \times b^4$, which corresponds to choice 2.