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

Question

Which of the expressions is equivalent to the expression?

6mn+3nm+9n2 6mn+\frac{3n}{m}+9n^2

Video Solution

Step-by-Step Solution

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

  • Step 1: Identify the common factor in the expression 6mn+3nm+9n2 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 6mn , 3nm\frac{3n}{m}, and 9n2 9n^2 , the common factor among them is clearly 3n 3n since:

  • 6mn 6mn can be divided by 3n 3n , giving 2m 2m .
  • 3nm \frac{3n}{m} can be divided by 3n 3n , giving 1m\frac{1}{m}.
  • 9n2 9n^2 can be divided by 3n 3n , giving 3n 3n .

Step 2: Factor Out the Common Factor

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

6mn+3nm+9n2=3n(2m)+3n(1m)+3n(3n) 6mn + \frac{3n}{m} + 9n^2 = 3n(2m) + 3n\left(\frac{1}{m}\right) + 3n(3n) .

This simplifies to:

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

Step 3: Compare with Choices

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

Therefore, the expression 6mn+3nm+9n2 6mn+\frac{3n}{m}+9n^2 is equivalent to 3n(2m+1m+3n) 3n(2m+\frac{1}{m}+3n) .

Answer

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