Breaking Down the Given Expressions

Oct 13, 2025 by pynkocean.com 0 views

Understanding the Concept of Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is a mathematical concept that represents the largest expression that divides each of the given expressions without leaving a remainder. In other words, it is the product of the common factors of the given expressions. The GCF is an essential concept in algebra, and it plays a crucial role in simplifying complex expressions.

Breaking Down the Given Expressions

To find the GCF of $16s^3t$ , $40s^5$ , and $68t^2$ , we need to break down each expression into its prime factors. This will help us identify the common factors and determine the GCF.

Prime Factorization of $16s^3t$

The prime factorization of $16s^3t$ is:

16s^3t = 2^4 \cdot s^3 \cdot t

Prime Factorization of $40s^5$

The prime factorization of $40s^5$ is:

40s^5 = 2^3 \cdot 5 \cdot s^5

Prime Factorization of $68t^2$

The prime factorization of $68t^2$ is:

68t^2 = 2^2 \cdot 17 \cdot t^2

Identifying the Common Factors

Now that we have the prime factorization of each expression, we can identify the common factors. The common factors are the factors that appear in all three expressions.

The common factors of $16s^3t$ , $40s^5$ , and $68t^2$ are:

$2^2$
$t$

Determining the Greatest Common Factor (GCF)

The GCF is the product of the common factors. In this case, the GCF is:

\text{GCF} = 2^2 \cdot t = 4t

However, we need to consider the exponents of the variables $s$ and $t$ . The expression with the lowest exponent of $s$ is $16s^3t$ , which has an exponent of $3$ . Therefore, the GCF should include $s^3$ .

The final answer is: $\boxed{4s^3t}$

Q: What is the Greatest Common Factor (GCF)?

A: The Greatest Common Factor (GCF) is a mathematical concept that represents the largest expression that divides each of the given expressions without leaving a remainder. It is the product of the common factors of the given expressions.

Q: How do I find the GCF of two or more expressions?

A: To find the GCF of two or more expressions, you need to follow these steps:

Break down each expression into its prime factors.
Identify the common factors among the expressions.
Determine the product of the common factors, which is the GCF.

Q: What is the difference between GCF and LCM?

A: The Greatest Common Factor (GCF) and Least Common Multiple (LCM) are two related but distinct mathematical concepts. The GCF is the largest expression that divides each of the given expressions without leaving a remainder, while the LCM is the smallest expression that is a multiple of each of the given expressions.

Q: How do I use the GCF to simplify complex expressions?

A: The GCF can be used to simplify complex expressions by factoring out the common factors. This can help to reduce the complexity of the expression and make it easier to work with.

Q: Can the GCF be a variable?

A: Yes, the GCF can be a variable. For example, if you have two expressions that contain the variable $x$ , the GCF may be a function of $x$ .

Q: How do I find the GCF of expressions with different bases?

A: To find the GCF of expressions with different bases, you need to follow these steps:

Convert each expression to its prime factorization.
Identify the common factors among the expressions.
Determine the product of the common factors, which is the GCF.

Q: Can the GCF be a constant?

A: Yes, the GCF can be a constant. For example, if you have two expressions that contain only constants, the GCF may be a constant.

Q: How do I use the GCF to solve equations?

A: The GCF can be used to solve equations by factoring out the common factors. This can help to simplify the equation and make it easier to solve.

Q: Can the GCF be a function of multiple variables?

A: Yes, the GCF can be a function of multiple variables. For example, if you have two expressions that contain multiple variables, the GCF may be a function of those variables.

Q: How do I find the GCF of expressions with negative coefficients?

A: To find the GCF of expressions with negative coefficients, you need to follow these steps:

Factor out the negative coefficients.
Identify the common factors among the expressions.
Determine the product of the common factors, which is the GCF.

Q: Can the GCF be a rational expression?

A: Yes, the GCF can be a rational expression. For example, if you have two expressions that contain rational numbers, the GCF may be a rational expression.

Q: How do I use the GCF to simplify rational expressions?

A: The GCF can be used to simplify rational expressions by factoring out the common factors. This can help to reduce the complexity of the expression and make it easier to work with.

Q: Can the GCF be a complex number?

A: Yes, the GCF can be a complex number. For example, if you have two expressions that contain complex numbers, the GCF may be a complex number.

Q: How do I find the GCF of expressions with complex coefficients?

A: To find the GCF of expressions with complex coefficients, you need to follow these steps:

Factor out the complex coefficients.
Identify the common factors among the expressions.
Determine the product of the common factors, which is the GCF.

Q: Can the GCF be a function of complex variables?

A: Yes, the GCF can be a function of complex variables. For example, if you have two expressions that contain complex variables, the GCF may be a function of those variables.

Q: How do I use the GCF to simplify complex expressions with complex coefficients?

A: The GCF can be used to simplify complex expressions with complex coefficients by factoring out the common factors. This can help to reduce the complexity of the expression and make it easier to work with.