In the expressions, x and y represent positive integers.expression one: 2 • 10xexpression two: 4 • 10x + y

To analyze and compare the two expressions given, let's break down the problem.

1. Break Down the Problem

We have two expressions:

• Expression One: $2 \cdot 10^x$
• Expression Two: $4 \cdot 10^x + y$

2. Relevant Concepts

The expressions involve simple algebraic manipulation of integers multiplied by powers of ten. Since $x$ and $y$ are positive integers, we know that:

• $10^x$ will always yield a positive outcome.
• $2 \cdot 10^x$ and $4 \cdot 10^x$ imply direct multiplication effects on the base of ten.

3. Analysis and Detail

• In Expression One, when you multiply $2$ by $10^x$, you get a certain number that is $2$ followed by $x$ zeros.
• In Expression Two, $4 \cdot 10^x$ is similar, yielding $4$ followed by $x$ zeros, and then we add $y$, which is a positive integer.

4. Verify and Summarize

To simplify the comparison:

• If $4 \cdot 10^x + y$ is compared to $2 \cdot 10^x$:
  • Note that $4 \cdot 10^x$ is clearly twice $2 \cdot 10^x$, so even without $y$, Expression Two is already larger.
  • The addition of $y$, which is positive, ensures that Expression Two is always greater than Expression One for all positive values of integers $x$ and $y$.

Final Answer

Thus, $4 \cdot 10^x + y > 2 \cdot 10^x$ for all positive integers $x$ and $y$.