How do you find the greatest common factor (GCF) of 4xy² and 20x²y⁴?

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The greatest common factor (GCF) is the largest expression that divides two or more expressions without leaving a remainder. To find the GCF of the expressions 4xy² and 20x²y⁴, we can follow these steps:

  1. Factor each term:
    • For 4xy²: Start by identifying the coefficients and variables.
      • Coefficient: 4
      • Variables: x and y (with the exponent of y being 2)
    • For 20x²y⁴: Similarly, we break this down.
      • Coefficient: 20
      • Variables: x (with the exponent of 2) and y (with the exponent of 4)
  2. Find the GCF of the coefficients:
    • The coefficients are 4 and 20.
      • Factors of 4: 1, 2, 4
      • Factors of 20: 1, 2, 4, 5, 10, 20

      The common factors are 1, 2, and 4, so the GCF of the coefficients is 4.

  3. Find the GCF of the variables:
    • For variable x: The smallest exponent is 1 (from 4xy²), so we take x¹.
    • For variable y: The smallest exponent is 2 (from 4xy²), so we take y².

Putting it all together, the GCF of 4xy² and 20x²y⁴ is the product of the GCFs of the coefficients and variables:

GCF = 4 * x¹ * y² = 4xy²

Thus, the greatest common factor of 4xy² and 20x²y⁴ is 4xy².

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