Find the LCD 3/(2x+10) and 1/(6x)
The question is asking to find the Lowest Common Denominator (LCD) for the two given rational expressions 3/(2x+10) and 1/(6x). Finding the LCD is a process used to determine the smallest common multiple of the denominators of two or more fractions or rational expressions. This common denominator is used to add, subtract, or compare fractions or rational expressions with different denominators. The challenge here lies in finding a single expression that both (2x+10) and (6x) can divide into without leaving a remainder, which would then allow for the addition or subtraction of the two given rational expressions.
$\frac{3}{2 x + 10}$and$\frac{1}{6 x}$
Answer
Solution:
Step 1:
Extract the factor of $2$ from the expression $2x + 10$.
Step 1.1:
Take out $2$ from $2x$ to get $\frac{3}{2(x) + 10}, \frac{1}{6x}$.
Step 1.2:
Extract $2$ from $10$ to obtain $\frac{3}{2x + 2 \cdot 5}, \frac{1}{6x}$.
Step 1.3:
Pull out $2$ from the entire expression $2x + 2 \cdot 5$ to simplify to $\frac{3}{2(x + 5)}, \frac{1}{6x}$.
Step 2:
The process of finding the LCD of several fractions is equivalent to determining the LCM of their denominators, which are $2(x + 5)$ and $6x$.
Step 3:
To find the LCM of $2(x + 5)$ and $6x$, which include both numbers and variables, follow these steps:
Determine the LCM of the numerical parts: $2$ and $6$.
Determine the LCM of the variable part: $x^1$.
Determine the LCM of the compound variable part: $x + 5$.
Multiply these LCMs together.
Step 4:
The LCM is the smallest number into which all the numbers can be divided without a remainder.
Enumerate the prime factors of each number.
Multiply each factor by the maximum number of times it appears in any of the numbers.
Step 5:
Since $2$ is a prime number, its factors are only $1$ and $2$.
Step 6:
The factors of $6$ are $2$ and $3$, which is $2 \cdot 3$.
Step 7:
Combine $2$ and $3$ to get $6$.
Step 8:
The variable $x^1$ is simply $x$, which appears once.
Step 9:
The LCM of $x^1$ is found by multiplying all prime factors by the greatest frequency they appear in any term, which is $x$.
Step 10:
The compound variable $x + 5$ is a factor in itself and occurs once.
Step 11:
The LCM of $x + 5$ is obtained by multiplying all factors by the maximum frequency they appear in any term, which is $x + 5$.
Step 12:
The Least Common Multiple (LCM) of a set of numbers is the smallest multiple that is divisible by each of the numbers, which in this case is $6x(x + 5)$.
Knowledge Notes:
The Least Common Denominator (LCD) is used to find a common denominator for fractions so that they can be compared or combined. It is the least common multiple (LCM) of the denominators.
Factoring is the process of breaking down expressions into products of simpler expressions. In this case, we factor out the greatest common factor (GCF), which is $2$.
The LCM of two or more numbers is the smallest number that is a multiple of all the numbers. It is useful for adding, subtracting, or comparing fractions.
Prime factorization is the process of breaking down a number into its prime factors. This is used to find the LCM by multiplying the highest powers of all prime factors involved.
Variables are treated similarly to numbers when finding the LCM. Each distinct variable or compound variable (like $x + 5$) is considered as a separate factor.
The LCM of variables is the variable raised to the highest power that occurs in any of the terms.
