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Chapter 6: Problem 12

Find the greatest common factor for each list of terms. \(a^{4} b^{5}, a^{3} b\)

Short Answer

Expert verified
The GCF is \(a^{3} b\).

Step by step solution

01

Identify the common base variables

The terms given are \(a^{4} b^{5}\) and \(a^{3} b\). Identify the common base variables in both terms, which are \(a\) and \(b\).
02

Determine the lowest power of each common base

For the base \(a\), the lowest power is 3 (since one term has \(a^{3}\) and the other has \(a^{4}\)). For the base \(b\), the lowest power is 1 (since one term has \(b\) and the other has \(b^{5}\)).
03

Construct the greatest common factor

Combine the lowest powers of each common base to construct the greatest common factor. For the base \(a\), use \(a^{3}\). For the base \(b\), use \(b^{1}\) or just \(b\).
04

Write the greatest common factor

The greatest common factor (GCF) of \(a^{4} b^{5}\) and \(a^{3} b\) is \(a^{3} b\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

base variables
Base variables are the fundamental building blocks of algebraic terms. In our example, we are looking at the terms \(a^{4} b^{5}\) and \(a^{3} b\). The base variables are the letters or symbols that are raised to some power in algebraic expressions. By identifying these base variables, we can begin to understand how the terms are constructed and how they relate to each other.

In our case, the base variables are \(a\) and \(b\). These are the variables that appear in both terms. Recognizing the common base variables is the first step in finding the greatest common factor (GCF).
lowest power
The lowest power is key when determining the GCF of algebraic terms with common base variables. The power of a base variable is indicated by the exponent attached to it. For example, in the term \(a^{4}\), the base variable \(a\) is raised to the power of 4. Similarly, in \(b^{5}\), the base variable \(b\) is raised to the power of 5.

To find the GCF, we look for the lowest power of each common base variable. In our example:

  • For the base \(a\), the terms are \(a^{4}\) and \(a^{3}\). The lowest power here is 3.
  • For the base \(b\), the terms are \(b^{5}\) and \(b\). The lowest power is 1.

Combining these lowest powers results in the GCF of the terms.
algebraic terms
Algebraic terms are expressions that consist of numbers, variables, and exponents. Each algebraic term is made up of a coefficient (a numeric part) and variables that may have exponents. They can be added or multiplied to form more complex expressions. Understanding the structure of algebraic terms helps in simplifying expressions and solving equations.

In our example, the algebraic terms \(a^{4} b^{5}\) and \(a^{3} b\) share common base variables. To find the GCF, we:

  • Identify the base variables \(a\) and \(b\).
  • Determine the lowest power of each common base variable.
  • Combine these lowest powers to form the GCF, which is \(a^{3} b\) in this case.

Grasping these concepts allows us to break down and understand more complex algebraic expressions effectively.

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Most popular questions from this chapter

Factor each polynomial. $$ y^{3} z+y^{2} z^{2}-6 y z^{3} $$

Find each product. $$ (3 a+2)(2 a+1) $$

Solve each equation, and check your solutions. $$ 6 x^{2}(2 x+3)=4(2 x+3)+5 x(2 x+3) $$

Factor completely. If the polynomial cannot be factored, write prime. $$ x^{2}+3 x-39 $$

Factor each trinomial completely. $$ 9 t^{2}+24 t r+16 r^{2} $$
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