In algebra, determining the **Greatest Common Factor (GCF)** of literal terms is a fundamental skill. The GCF of two terms is the highest expression that divides each of the terms without leaving a remainder. When dealing with variables, the GCF is found by identifying the smallest exponent for each common variable in the terms. This article walks you through the process of finding the GCF of the literal terms **m ^{7}n^{4}p^{3} and mn^{12}p^{5}**.

### Step 1: Factor Each Term into Its Variable Components

Begin by expressing each term as a product of variables raised to their respective exponents.

**First Term:**m^{7}n^{4}p^{3}*m*: Variable^{7}*m*raised to the 7th power.*n*: Variable^{4}*n*raised to the 4th power.*p*: Variable^{3}*p*raised to the 3rd power.

**Second Term:**mn^{12}p^{5}- m: Variable
*m*raised to the 1st power (since no exponent is shown). - n
^{12}: Variable*n*raised to the 12th power. - p
^{5}: Variable*p*raised to the 5th power.

- m: Variable

### Step 2: Identify Common Variables

List the variables that appear in both terms. In this case, all three variables *m, n, *and *p* are present in both terms.

### Step 3: Determine the GCF for Each Common Variable

For each common variable, the GCF will include the variable raised to the **smallest exponent** found in the two terms.

**Variable m:**- Exponents: 7 (in m
^{7}) and 1 (in m) **GCF Component:**m^{1}=m

- Exponents: 7 (in m
**Variable n:**- Exponents: 4 (in n
^{4}) and 12 (in n^{12}) **GCF Component:**n^{4}

- Exponents: 4 (in n
**Variable p:**- Exponents: 3 (in p
^{3}) and 5 (in p^{5}) **GCF Component:**p^{3}

- Exponents: 3 (in p

### Step 4: Combine the GCF Components

Multiply the GCF components of each variable to obtain the overall GCF of the two terms.

**GCF** = m×n^{4}×p^{3} = mn^{4}p^{3}

### Answer:

The **Greatest Common Factor (GCF)** of the literal terms ** m^{7}n^{4}p^{3} and mn^{12}p^{5} **is:

*mn*^{4}p^{3}This GCF represents the highest expression that evenly divides both original terms, ensuring that no larger common factor exists. Understanding this process is essential for simplifying algebraic expressions, factoring polynomials, and solving various algebraic equations.