Problem 9 - Algebra A Combined Function

Question

Find the \(G C F\) for each list. $$ z^{7}, z^{9}, z^{11} $$

Step-by-Step Solution

Answer

The GCF is \(z^7\).

Step 1: Understand the Problem

The goal is to find the greatest common factor (GCF) of the given monomials: \(z^7, z^9, z^{11}\), which represent powers of the variable \(z\).

Step 2: Identify the Exponents

For each term, identify the exponent of the variable \(z\): \(z^7\) has an exponent of 7, \(z^9\) has an exponent of 9, and \(z^{11}\) has an exponent of 11.

Step 3: Find the Smallest Exponent

The GCF for terms that are powers of a single variable is the smallest exponent. Among the exponents 7, 9, and 11, the smallest is 7.

Step 4: Write the GCF

The GCF of \(z^7, z^9, z^{11}\) is \(z^7\), which is the power of the variable with the smallest exponent among the given terms.

Key Concepts

Monomials

Monomials are mathematical expressions that consist of a single term. They can include constants, variables, or the product of both. For example, in the expression \(z^7\), \(z\) is the variable and 7 is the exponent. Monomials typically do not involve any addition or subtraction. Hence, they are straightforward to work with. Understanding monomials sets the foundation for working with polynomials, which are combinations of multiple monomials. When dealing with problems like finding the greatest common factor (GCF) among monomials, we first identify the individual elements in each monomial term. These include:

• The coefficient, if any, like numbers in front of variables (e.g., 3 in \(3x^2\))
• The variable itself (e.g., \(z\) in \(z^7\))
• The exponent, which denotes the power to which the variable is raised

Mastering monomials is essential for solving more complex algebraic expressions and equations effectively.

Exponents

Exponents are foundational in algebra, indicating how many times a number or variable is multiplied by itself. In the expression \(z^7\), 7 is the exponent. This signifies that \(z\) is multiplied by itself 7 times. Understanding exponents is crucial as they help simplify expressions and solve equations efficiently. Exponents follow specific laws or rules, such as:

• Product of Powers: When multiplying like bases, you add the exponents (e.g., \(z^m \times z^n = z^{m+n}\)).
• Power of a Power: When raising a power to another power, you multiply the exponents (e.g., \((z^m)^n = z^{m \cdot n}\)).
• Quotient of Powers: When dividing like bases, you subtract the exponents (e.g., \(z^m/z^n = z^{m-n}\)).

Using these rules helps in simplifying expressions, making it easier to work with powers and identify factors.

Powers of Variables

Powers of variables occur when a variable is raised to an exponent, as seen in monomials like \(z^7\). These powers aid in describing repeated multiplication and are particularly important in algebraic operations.
The greatest common factor (GCF) of powers of a variable involves looking at the exponents. For terms with a common variable, the GCF is typically the term with the smallest exponent. For example, among \(z^7\), \(z^9\), and \(z^{11}\), the smallest exponent is 7, making \(z^7\) the GCF. Identifying the smallest exponent when finding the GCF is crucial, as it ensures the factor found is indeed present in all terms being compared. This method streamlines the process without needing to multiply or expand entire expressions. By focusing on the exponents, students can determine the simplifying factor more efficiently.