Find the LCD (m+n)/(12mn) and (n+m)/(18m^3n)

$\frac{m + n}{12 m n}$ and $\frac{n + m}{18 m^{3} n}$

Step 1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$12 m n, 18 m^{3} n$

Step 2

Since $12 m n, 18 m^{3} n$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $12, 18$ then find LCM for the variable part $m^{1}, n^{1}, m^{3}, n^{1}$ .

Step 3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 4

The prime factors for $12$ are $2 \cdot 2 \cdot 3$ .

$12$ has factors of $2$ and $6$ .

$2 \cdot 6$

$6$ has factors of $2$ and $3$ .

$2 \cdot 2 \cdot 3$

Step 5

The prime factors for $18$ are $2 \cdot 3 \cdot 3$ .

$18$ has factors of $2$ and $9$ .

$2 \cdot 9$

$9$ has factors of $3$ and $3$ .

$2 \cdot 3 \cdot 3$

Step 6

The LCM of $12, 18$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 3 \cdot 3$

Step 7

Multiply $2 \cdot 2 \cdot 3 \cdot 3$ .

Multiply $2$ by $2$ .

$4 \cdot 3 \cdot 3$

Multiply $4$ by $3$ .

$12 \cdot 3$

Multiply $12$ by $3$ .

$36$

Step 8

The factor for $m^{1}$ is $m$ itself.

$m^{1} = m$

$m$ occurs $1$ time.

Step 9

The factor for $n^{1}$ is $n$ itself.

$n^{1} = n$

$n$ occurs $1$ time.

Step 10

The factors for $m^{3}$ are $m \cdot m \cdot m$ , which is $m$ multiplied by each other $3$ times.

$m^{3} = m \cdot m \cdot m$

$m$ occurs $3$ times.

Step 11

The factor for $n^{1}$ is $n$ itself.

$n^{1} = n$

$n$ occurs $1$ time.

Step 12

The LCM of $m^{1}, n^{1}, m^{3}, n^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$m \cdot m \cdot m \cdot n$

Step 13

Simplify $m \cdot m \cdot m \cdot n$ .

Multiply $m$ by $m$ .

$m^{2} \cdot m \cdot n$

Multiply $m^{2}$ by $m$ by adding the exponents.

Multiply $m^{2}$ by $m$ .

Raise $m$ to the power of $1$ .

$m^{2} \cdot m^{1} \cdot n$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$m^{2 + 1} \cdot n$

Add $2$ and $1$ .

$m^{3} \cdot n$

$m^{3} n$

Step 14

The LCM for $12 m n, 18 m^{3} n$ is the numeric part $36$ multiplied by the variable part.

$36 m^{3} n$

Find the LCD (m+n)/(12mn) and (n+m)/(18m^3n)

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