Mathematical Value Comparison: Greater Than Analysis

To determine which of the given expressions is the greatest, we will use the relevant exponent rules to simplify each one:

• Simplify $y^7 \times y^2$:
  Using the Product of Powers rule, we have $y^7 \times y^2 = y^{7+2} = y^9$.
• Simplify $(y^4)^3$:
  Using the Power of a Power rule, we have $(y^4)^3 = y^{4 \times 3} = y^{12}$.
• Simplify $y^9$:
  This expression is already simplified and is $y^9$.
• Simplify $\frac{y^{11}}{y^4}$:
  Using the Division of Powers rule, we have $\frac{y^{11}}{y^4} = y^{11-4} = y^7$.

After simplifying, we compare the powers of $y$ from each expression:

• $y^9$ from $y^7 \times y^2$
• $y^{12}$ from $(y^4)^3$
• $y^9$ from $y^9$
• $y^7$ from $\frac{y^{11}}{y^4}$

Clearly, $y^{12}$ is the largest power among the expressions, meaning that $(y^4)^3$ is the greatest value.

Therefore, the correct choice is $(y^4)^3$.