Finite Math Examples

$\frac{8}{7 a b} - \frac{3}{14 a^{2} b} + \frac{5}{21 a b^{2}}$

Step 1

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

$7 a b, 14 a^{2} b, 21 a b^{2}$

Step 2

Since $7 a b, 14 a^{2} b, 21 a b^{2}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $7, 14, 21$ then find LCM for the variable part $a^{1}, b^{1}, a^{2}, b^{1}, a^{1}, b^{2}$ .

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

Since $7$ has no factors besides $1$ and $7$ .

$7$ is a prime number

Step 5

$14$ has factors of $2$ and $7$ .

$2 \cdot 7$

Step 6

$21$ has factors of $3$ and $7$ .

$3 \cdot 7$

Step 7

The LCM of $7, 14, 21$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 3 \cdot 7$

Step 8

Multiply $2 \cdot 3 \cdot 7$ .

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Step 8.1

Multiply $2$ by $3$ .

$6 \cdot 7$

Step 8.2

Multiply $6$ by $7$ .

$42$

Step 9

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

$a^{1} = a$

$a$ occurs $1$ time.

Step 10

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

$b^{1} = b$

$b$ occurs $1$ time.

Step 11

The factors for $a^{2}$ are $a \cdot a$ , which is $a$ multiplied by each other $2$ times.

$a^{2} = a \cdot a$

$a$ occurs $2$ times.

Step 12

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

$b^{1} = b$

$b$ occurs $1$ time.

Step 13

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

$a^{1} = a$

$a$ occurs $1$ time.

Step 14

The factors for $b^{2}$ are $b \cdot b$ , which is $b$ multiplied by each other $2$ times.

$b^{2} = b \cdot b$

$b$ occurs $2$ times.

Step 15

The LCM of $a^{1}, b^{1}, a^{2}, b^{1}, a^{1}, b^{2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$a \cdot a \cdot b \cdot b$

Step 16

Simplify $a \cdot a \cdot b \cdot b$ .

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Step 16.1

Multiply $a$ by $a$ .

$a^{2} \cdot b \cdot b$

Step 16.2

Multiply $b$ by $b$ by adding the exponents.

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Step 16.2.1

Move $b$ .

$a^{2} \cdot (b \cdot b)$

Step 16.2.2

Multiply $b$ by $b$ .

$a^{2} \cdot b^{2}$

$a^{2} b^{2}$

Step 17

The LCM for $7 a b, 14 a^{2} b, 21 a b^{2}$ is the numeric part $42$ multiplied by the variable part.

$42 a^{2} b^{2}$