Find the LCD 3/(2x+10) and 1/(6x)

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:

1. Determine the LCM of the numerical parts: $2$ and $6$.

2. Determine the LCM of the variable part: $x^1$.

3. Determine the LCM of the compound variable part: $x+5$.

4. Multiply these LCMs together.

Step 4:

The LCM is the smallest number into which all the numbers can be divided without a remainder.

1. Enumerate the prime factors of each number.

2. 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.