Problem 12 - Algebra A Combined Function

Question

Find the \(G C F\) for each list. $$ p^{7} q, p^{8} q^{2}, p^{9} q^{3} $$

Step-by-Step Solution

Answer

The GCF is \( p^7 q \).

Step 1: Identify the Common Terms

The given terms are \( p^7 q, p^8 q^2, p^9 q^3 \). We need to determine the greatest common factor (GCF) for these expressions by focusing on the variables \( p \) and \( q \) separately.

Step 2: Determine the Common Bases and Lower Exponents

For the variable \( p \), the bases are \( p^7, p^8, \) and \( p^9 \). The smallest exponent here is 7. For the variable \( q \), the bases are \( q, q^2, \) and \( q^3 \). The smallest exponent is 1.

Step 3: Construct the GCF from the Common Bases and Lowest Exponents

Combine the results from Step 2 to form the GCF. The common factor is derived by taking the variable with the lowest power. Thus, the GCF is \( p^7 q^1 \) or simply \( p^7 q \).

Key Concepts

Exponents

Exponents are a key mathematical concept that help in simplifying expressions with repeated multiplication. When you see something like \(p^7\), it means "\(p\) to the power of 7," which is the same as multiplying \(p\) by itself seven times.
Understanding exponents is crucial when working with algebraic expressions, as they allow you to manage larger numbers compactly.Some important properties of exponents include:

• **Product of Powers:** If you multiply numbers with the same base, you add their exponents, like \(p^a \times p^b = p^{a+b}\).
• **Quotient of Powers:** When you divide like bases, you subtract the exponents, shown as \(p^a / p^b = p^{a-b}\).
• **Power of a power:** If you raise a power to another power, you multiply the exponents, such as \((p^a)^b = p^{a \times b}\).

Learning to manage and simplify expressions with exponents is a critical skill, especially when identifying the greatest common factor (GCF) where you must identify the lowest exponent of shared bases.

Variables in Algebra

Variables in algebra are symbols, often letters, used to represent numbers in equations and expressions. They play a fundamental role in algebraic expressions, acting as placeholders for unknown values or quantities that can vary.
Let's use our exercise example: the letters \(p\) and \(q\) are variables. In the expressions \(p^7 q\), \(p^8 q^2\), and \(p^9 q^3\), variables are raised to different powers, showing their impact on the whole term.Key points about variables include:

• **Variables can change:** They are not fixed; their value can vary according to the context of the problem or expression.
• **Interaction with Operations:** Variables interact through algebraic operations like addition, subtraction, multiplication, and division, just like numbers.
• **Expressions Dictate Variables' Roles:** An expression determines how and where variables interact, such as showing relationships between quantities.

Understanding how to manipulate and work with variables allows you to simplify expressions and solve equations effectively. They are essential for identifying common factors in algebraic expressions by comparing coefficients and exponents.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division. They represent a phrase in algebra that demonstrates a mathematical relationship. In the given exercise, each expression like \(p^7 q\), \(p^8 q^2\), and \(p^9 q^3\) is an example of an algebraic expression.
These expressions often contain:

• **Constants:** Fixed numerical values which are combined with variables.
• **Coefficients:** Numbers that multiply variables in terms such as 3 in \(3x\).
• **Terms:** The parts of the expression separated by '+' or '-' signs.

With algebraic expressions, like those you see in the greatest common factor (GCF) exercise, it's crucial to break down terms and find common elements. This involves analyzing variables and coefficients, recognizing common divisors, and applying properties of exponents to simplify or compare the expressions.By grasping these basic components of algebraic expressions, you'll be equipped to tackle more complex algebraic problems and understand how to manipulate variables to find factors like the GCF.