[Solved] What is the GCF of 48m5n and 81m2n2?

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 48m⁵n and 81m²n².

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: 48m⁵n
  • Numerical Coefficient: 48
  • Variables:
    • m⁵n: Variable m raised to the 5th power.
    • n: Variable n raised to the 1st power (since no exponent is shown).
• Second Term: 81m²n²
  • Numerical Coefficient: 81
  • Variables:
    • m²: Variable m raised to the 2nd power.
    • n²: Variable n raised to the 2nd power.

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=2⁴×3¹

Prime Factorization of 81: 81=3×27=3×3×9=3×3×3×3=3⁴

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: 3¹

In 81: 3⁴

GCF Component: 3^min⁡(1,4)=3¹=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 m⁵) and 2 (in m²)
  • GCF Component: m^min⁡(5,2)=m²
• Variable n:
  • Exponents: 1 (in n) and 2 (in n²)
  • GCF Component: n^min⁡(1,2)=n¹=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×m²×n=3m²n

Answer:

The Greatest Common Factor (GCF) of the literal terms 48m⁵n and 81m²n² = 3m²n