Simplify the Expression: y^9 × y^2 × y^3 Using Power Properties

To solve the problem of simplifying the expression $y^9 \times y^2 \times y^3$, we will follow these steps:

First, recognize that the expression entails powers of the same base $y$, and we can use the rule for multiplying powers with the same base. This rule states that when multiplying like bases, we add the exponents. Mathematically, this can be expressed as:

• Step 1: Identify the base $y$ and the exponents $9$, $2$, and $3$.
• Step 2: Apply the rule for multiplication of powers $y^9 \times y^2 \times y^3 = y^{9+2+3}$.
• Step 3: Calculate the sum of the exponents: $9 + 2 + 3 = 14$.
• Step 4: Simplify the expression to yield the solution, $y^{14}$.

In reviewing the answer choices:

• Choice 1: $y^{9+2+3}$ represents the fully simplified expression, which agrees with our solution.
• Choice 2: $y^{9+2} \times y^3$ and choice 3: $y^9 \times y^{2+3}$ still maintain some intermediate steps of simplification. Yet, both can eventually be simplified further to $y^{14}$.

Therefore, all expressions represent correct approaches or intermediates toward achieving the correct final form. Thus, All answers are correct.