Problem 7 - Algebra A Combined Function

Question

Find the \(G C F\) for each list. $$ y^{2}, y^{4}, y^{7} $$

Step-by-Step Solution

Answer

The GCF is \( y^2 \).

Step 1: Identify the List of Terms

We have three terms in our list: \( y^2, y^4, \) and \( y^7 \). We need to find the greatest common factor (GCF) for these terms.

Step 2: Express Each Term as a Power of y

Each term is already expressed as a power of \( y \): \( y^2, y^4, \text{ and } y^7 \).

Step 3: Find the Smallest Exponent

To find the GCF of terms with the same base, we identify the smallest exponent from the given terms. Here, the exponents are 2, 4, and 7. The smallest exponent is 2.

Step 4: Write the GCF using the Smallest Exponent

Since the smallest exponent found in Step 3 is 2, the GCF is \( y^2 \).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. It is an essential concept in algebra and provides a simpler way to express repeated multiplication.

• The base is the number that is being multiplied.
• The exponent tells us how many times to use the base in multiplication.

For instance, in the expression \( y^4 \), \( y \) is the base, and \( 4 \) is the exponent. This tells us to multiply \( y \) by itself three more times, resulting in \( y \times y \times y \times y \). This concept helps simplify expressions and is foundational for understanding larger algebraic structures.

Algebra

Algebra is a branch of mathematics that uses symbols to represent numbers in equations and expressions. It allows us to solve problems where quantities are unknown by setting up equations.

• Variables like \( y \) in \( y^7 \) can vary in value and are used to represent unknown quantities.
• Expressions combine numbers and variables, such as \( y^2 \), to form meaningful mathematical statements.

Understanding algebra is key to solving various mathematical problems, including finding the greatest common factor (GCF). By using variables and exponents, algebra helps us systematically approach and solve complex numeric problems.

Finding GCF

The greatest common factor (GCF) is the largest factor that divides all numbers in a given set without leaving a remainder. When working with algebraic terms, the GCF is derived by identifying the smallest common term or expression.

• First, identify the terms: \( y^2, y^4, \) and \( y^7 \) are the terms given in our exercise.
• Determine the smallest exponent (here it is 2 from \( y^2 \)), as all terms share the base \( y \).
• The GCF is the term with the smallest exponent, \( y^2 \).

Finding the GCF is helpful in simplifying expressions and solving equations, ensuring mathematical expressions are easily manageable. This process is essential in factoring polynomials, reducing fractions, and finding common denominators.