Determining the **Greatest Common Factor (GCF)** of algebraic expressions involves identifying the highest expression that divides each term without leaving a remainder. When dealing with both numerical coefficients and variables, the process encompasses finding the GCF of the numerical parts and the lowest exponents of the common variables. This article provides a comprehensive solution to finding the GCF of the literal terms ** 48m^{5}n and 81m^{2}n^{2}**.

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

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

**First Term:**48m^{5}n**Numerical Coefficient:**48**Variables:**- m
^{5}n: Variable m raised to the 5th power. - n: Variable n raised to the 1st power (since no exponent is shown).

- m

**Second Term:**81m^{2}n^{2}**Numerical Coefficient:**81**Variables:**- m
^{2}: Variable*m*raised to the 2nd power. - n
^{2}: Variable*n*raised to the 2nd power.

- m

## Step 2: Find the GCF of the Numerical Coefficients

To determine the GCF of the numerical parts (48 and 81), perform prime factorization on each number.

**Prime Factorization of 48:** 48=2×24=2×2×12=2×2×2×6=2×2×2×2×3=2^{4}×3^{1}

**Prime Factorization of 81:** 81=3×27=3×3×9=3×3×3×3=3^{4}

**Identify Common Prime Factors:** The only common prime factor between 48 and 81 is 3.

**Determine the Lowest Exponent:** **For prime number 3:**

**In 48:** 3^{1}

**In 81:** 3^{4}

**GCF Component:** 3^{min(1,4)}=3^{1}=3

**GCF of Numerical Coefficients:** GCF_{numerical}=3

## Step 3: Identify Common Variables and Determine Their GCF

For the variables, the GCF is found by taking the variable with the lowest exponent present in both terms.

**Variable m:****Exponents:**5 (in m^{5}) and 2 (in m^{2})**GCF Component:**m^{min(5,2)}=m^{2}

**Variable n:****Exponents:**1 (in n) and 2 (in n^{2})**GCF Component:**n^{min(1,2)}=n^{1}=n

### Step 4: Combine the GCF Components

Multiply the GCF of the numerical coefficients with the GCF of the variable components to obtain the overall GCF.

**GCF=GCF _{numerical}×GCF_{m}×GCF_{n}=3×m^{2}×n=3m^{2}n**

### Answer:

The **Greatest Common Factor (GCF)** of the literal terms 48m^{5}n and 81m^{2}n^{2} = **3m ^{2}n**