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what is the gcf of h4 and h8

what is the gcf of h4 and h8

2 min read 05-02-2025
what is the gcf of h4 and h8

Finding the greatest common factor (GCF) of algebraic expressions like h⁴ and h⁸ is a fundamental concept in algebra. This article will explain how to determine the GCF of h⁴ and h⁸, and provide a broader understanding of the process.

Understanding Greatest Common Factor (GCF)

The greatest common factor (GCF), also known as the greatest common divisor (GCD), is the largest number that divides evenly into two or more numbers. In algebra, this extends to variables and exponents.

The GCF represents the largest expression that can be factored out from a set of terms. This is crucial in simplifying expressions and solving equations.

Finding the GCF of h⁴ and h⁸

Let's break down how to find the GCF of h⁴ and h⁸.

1. Prime Factorization (Not Strictly Necessary Here, but Helpful for Understanding):

While not strictly necessary for this simple example, understanding prime factorization helps solidify the concept. We can represent h⁴ and h⁸ as:

  • h⁴ = h * h * h * h
  • h⁸ = h * h * h * h * h * h * h * h

Notice that both expressions are composed entirely of the variable 'h'.

2. Identifying Common Factors:

The common factor is the variable 'h'. We need to determine the highest power of 'h' that divides both h⁴ and h⁸.

3. Determining the Highest Common Power:

Compare the exponents: 4 and 8. The smallest exponent is 4. Therefore, the highest power of 'h' that divides both terms is h⁴.

Conclusion:

The greatest common factor (GCF) of h⁴ and h⁸ is h⁴. This means h⁴ is the largest expression that can divide both h⁴ and h⁸ without leaving a remainder.

Expanding the Concept: GCF with Coefficients and Multiple Variables

The process extends to more complex expressions. For example, let's consider finding the GCF of 6x²y³ and 18x⁴y.

  1. Find the GCF of the coefficients: The GCF of 6 and 18 is 6.
  2. Find the GCF of the x terms: The GCF of x² and x⁴ is x².
  3. Find the GCF of the y terms: The GCF of y³ and y is y.

Therefore, the GCF of 6x²y³ and 18x⁴y is 6x²y.

Practice Problems

Try finding the GCF for these expressions:

  • 12a³b² and 18a²b⁴
  • 25m⁵n² and 15m³n³
  • x⁶ and x¹²

Remember, finding the GCF is essential for simplifying expressions and solving various algebraic problems. Mastering this concept builds a strong foundation in algebra.

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