Question

For Exercises, find the GCF of the monomials. \(56 x^{9}\) and \(72 x^{7}\)

Short answer

Question: Determine the Greatest Common Factor (GCF) of the monomials \(56 x^{9}\) and \(72 x^{7}\). Answer: The GCF of the monomials \(56x^9\) and \(72x^7\) is \(8x^{7}\).

Step-by-step solution

Determine the GCF of the coefficients

To find the GCF of the numbers 56 and 72, first list all the factors of each number and then identify the largest number that appears on both lists. The factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. The largest number that appears in both lists is 8, so the GCF of the coefficients is 8.

Determine the lowest exponent of the variables

Since both monomials have the same variable (x), we need to find the smallest exponent of x between the two terms. The exponents are 9 and 7, and the smallest exponent is 7. Therefore, the lowest exponent for the variable x is \(x^{7}\).

Combine the GCF of coefficients and the lowest exponent of the variables

Now that we have found the GCF of the coefficients (8) and the lowest exponent of the variables (\(x^{7}\)), combine them to get the final answer. The GCF of the monomials \(56x^9\) and \(72x^7\) is \(8x^{7}\).

Key concepts

Monomials

In mathematics, a monomial is a single term algebraic expression. It consists of three main parts: a coefficient, variables, and the exponents of those variables. Understanding monomials is essential for many types of calculations, including determining the greatest common factor (GCF). A monomial can look simple like

• \(x \)
• \(5xy \)
• \(6a^3b^2 \)

Typically, monomials are used in expressions where individual terms are combined to form polynomials. When finding the GCF of monomials, it requires determining the common factors of these components. In other words, we're looking for the largest algebraic form that divides each term without a remainder.

To better grasp monomials, visualize them as building blocks in algebra, which can be pieced together to form bigger expressions. By practicing with simple monomials, students become adept at identifying coefficients and exponents needed for solving more complex algebraic problems.

Coefficients

Coefficients in a monomial are the numerical part. They represent how much of a certain variable we have, serving as the scale factor multiplied by the variable. For instance, in the monomial \(56x^9\), the coefficient is 56. To solve problems involving monomials, you must understand how to manipulate these numerical parts.

• Identify the coefficient in a given monomial.
• Factor each coefficient to find common divisors.
• Determine the largest common factor, which is the GCF of the coefficients.

In the example given in the original problem, the coefficients are 56 and 72. By listing the factors of each and identifying the largest common factor, which is 8, we find the GCF of these coefficients.

Practicing with different sets of coefficients will enhance your ability to quickly identify the GCF, which is crucial in simplifying expressions and solving algebraic equations.

Exponents

Exponents in a monomial are the powers to which a variable is raised. They signify how many times the variable is multiplied by itself. In the term \(x^9\), 9 is the exponent. Exponents play a crucial role in defining how potent a monomial is compared to another one.

When dealing with monomials, identifying the smallest exponent is necessary to find the GCF. In our example, we have two exponents for the same variable: 9 and 7 (\(x^9\) and \(x^7\)). The smallest of these exponents is 7, leading to the lowest power of the variable in the GCF.

• Understand that only exponents of like variables are compared.
• Always use the smallest exponent value in the GCF.
• Realize that this process simplifies expressions by reducing them to their simplest form.

Mastering exponents is vital for algebra, as it helps in both simplifying and solving complex equations. Working through exercises involving different exponents strengthens comprehension and efficiency in handling power and degree of expressions.