Factor the Expression: Simplifying 27mn^3+33m^2n^5

To solve the problem of factoring the expression $27mn^3 + 33m^2n^5$, we'll follow these steps:

• Step 1: Find the greatest common factor (GCF) of the coefficients.
• Step 2: Determine the GCF of the variable terms.
• Step 3: Factor out the GCF from the original expression.

Now, let's work through each step:

Step 1: Look at the coefficients 27 and 33. The greatest common factor of 27 and 33 is 3. This is the largest number that divides both 27 and 33 without leaving a remainder.

Step 2: Consider the variables in each term:

• In $27mn^3$, the variable part is $mn^3$.
• In $33m^2n^5$, the variable part is $m^2n^5$.

For the variable $m$, the lowest power common to both terms is $m^1$.

For the variable $n$, the lowest power common is $n^3$.

Therefore, the greatest common factor (GCF) of the variable part is $mn^3$.

Step 3: Combine the GCF of the coefficients and the variable terms:

The GCF of the entire expression is $3mn^3$.

Now, divide each term of the expression by this GCF:

• Divide $27mn^3$ by $3mn^3$, which leaves $9$.
• Divide $33m^2n^5$ by $3mn^3$, which gives $11mn^2$.

Thus, the factored form of the expression is:

$3mn^3 (9 + 11mn^2)$

In conclusion, the solution to the problem is $3mn^3 (9 + 11mn^2)$.