Problem 14
Question
For the following problems, perform the divisions. $$ \frac{6 a+12}{2} $$
Step-by-Step Solution
Verified Answer
Answer: The simplified result of the division is \(3(a + 2)\).
1Step 1: Identify the numerator and denominator
The problem asks to find the result of the division of the given fraction:
$$
\frac{6a + 12}{2}
$$
The numerator is \(6a + 12\) and the denominator is \(2\).
2Step 2: Factor out the greatest common factor in the numerator
In the numerator, the greatest common factor is \(6\). Factor out \(6\) from the expression:
$$
6a + 12 = 6(a + 2)
$$
3Step 3: Divide each term in the numerator by the denominator
Now that the numerator is factored, perform the division by dividing each term with the denominator. Divide the coefficient \(6\) by the denominator \(2\):
$$
\frac{6(a + 2)}{2} = \frac{6}{2}(a + 2)
$$
Calculate the division:
$$
\frac{6}{2}(a + 2) = 3(a + 2)
$$
4Step 4: Simplify the result
The simplified result of the division is:
$$
3(a + 2)
$$
Key Concepts
Simplifying Algebraic ExpressionsFactoring Greatest Common FactorFraction Division
Simplifying Algebraic Expressions
Simplifying algebraic expressions is a fundamental skill in algebra that involves reducing an expression to its most basic form. This is done by performing operations like addition, subtraction, multiplication, and division, as well as combining like terms and factoring.
For example, take the expression \( \frac{6a + 12}{2} \). The primary goal is to simplify this expression so that it's easier to understand and work with. We start by identifying the terms that can be combined or factored. In this case, the terms in the numerator share a common factor, which we can factor out to simplify the expression.
Factoring out the common elements leads us to a more manageable form, which then allows us to perform the division with greater ease. The essence of simplifying an algebraic expression lies in making the equation cleaner and more direct, thus enhancing our ability to manipulate and solve it.
For example, take the expression \( \frac{6a + 12}{2} \). The primary goal is to simplify this expression so that it's easier to understand and work with. We start by identifying the terms that can be combined or factored. In this case, the terms in the numerator share a common factor, which we can factor out to simplify the expression.
Factoring out the common elements leads us to a more manageable form, which then allows us to perform the division with greater ease. The essence of simplifying an algebraic expression lies in making the equation cleaner and more direct, thus enhancing our ability to manipulate and solve it.
Factoring Greatest Common Factor
Factoring out the greatest common factor (GCF) is a process used to simplify expressions or solve equations efficiently. The GCF is the largest factor that divides two or more numbers or terms. By extracting the GCF, we reduce the complexity of the expression.
In our problem, \(6a + 12\), the GCF of the terms is \(6\). We factor out this GCF by dividing each term in the numerator by \(6\):\[ 6a + 12 = 6(a + 2) \]
It's important to recognize the GCF in algebraic expressions because it offers a path to simplification that can lead to easier addition, subtraction, or further factoring. Mastering this technique is vital for students to progress through more advanced algebra topics.
In our problem, \(6a + 12\), the GCF of the terms is \(6\). We factor out this GCF by dividing each term in the numerator by \(6\):\[ 6a + 12 = 6(a + 2) \]
It's important to recognize the GCF in algebraic expressions because it offers a path to simplification that can lead to easier addition, subtraction, or further factoring. Mastering this technique is vital for students to progress through more advanced algebra topics.
Fraction Division
The division of fractions involves simplifying a complex fraction into its simplest form. In algebra, when dividing terms or coefficients, we employ this concept to make an expression more straightforward.
In the provided exercise, we use fraction division to divide the coefficient of the factored expression by the denominator. Here's how it works mathematically:\[ \frac{6}{2}(a + 2) = 3(a + 2) \]
Simply put, we divide the numerator's coefficient by the denominator to get a new coefficient, which we then apply to the remaining expression within the parentheses. This step is crucial in the simplification process as it effectively reduces the fraction to its most reduced form, thus making it easier to interpret and solve related algebraic problems.
In the provided exercise, we use fraction division to divide the coefficient of the factored expression by the denominator. Here's how it works mathematically:\[ \frac{6}{2}(a + 2) = 3(a + 2) \]
Simply put, we divide the numerator's coefficient by the denominator to get a new coefficient, which we then apply to the remaining expression within the parentheses. This step is crucial in the simplification process as it effectively reduces the fraction to its most reduced form, thus making it easier to interpret and solve related algebraic problems.
Other exercises in this chapter
Problem 13
For the following problems, reduce each rational expression to lowest terms. $$ \frac{6}{6 x-18} $$
View solution Problem 14
$$ \frac{16 x^{5}\left(x^{2}-1\right)}{9 x-9} \div \frac{2 x^{2}-2 x}{3} $$
View solution Problem 14
For the following problems, fill in the missing term. $$ -\frac{3}{y-5}=\frac{\underline{\phantom{xx}}}{y-5} $$
View solution Problem 14
Four fifths of a number added to the reciprocal of number yields \(\frac{81}{10} .\) What is the number?
View solution