Simplify the Expression: 6mn + 3n/m + 9n²

Which of the expressions is equivalent to the expression?

$6mn+\frac{3n}{m}+9n^2$

To solve the problem, we'll follow these steps:

• Step 1: Identify the common factor in the expression $6mn+\frac{3n}{m}+9n^2$.
• Step 2: Factor this common factor out from the expression.
• Step 3: Compare the factored expression to the given choices.

Let's work through each step:

Step 1: Identify the Common Factor

Looking at the terms $6mn$, $\frac{3n}{m}$, and $9n^2$, the common factor among them is clearly $3n$ since:

• $6mn$ can be divided by $3n$, giving $2m$.
• $\frac{3n}{m}$ can be divided by $3n$, giving $\frac{1}{m}$.
• $9n^2$ can be divided by $3n$, giving $3n$.

Step 2: Factor Out the Common Factor

Factoring $3n$ out of each term, we rewrite the expression:

$6mn + \frac{3n}{m} + 9n^2 = 3n(2m) + 3n\left(\frac{1}{m}\right) + 3n(3n)$.

This simplifies to:

$3n(2m + \frac{1}{m} + 3n)$.

Step 3: Compare with Choices

We compare our factored expression, $3n(2m + \frac{1}{m} + 3n)$, to the given choices. We find that Choice 1 matches our factored form.

Therefore, the expression $6mn+\frac{3n}{m}+9n^2$ is equivalent to $3n(2m+\frac{1}{m}+3n)$.