Finite Math Examples

20 c d

$20 c d$ ,

40 a^{2} c^{2} d^{4}

$40 a^{2} c^{2} d^{4}$ ,

15 a b d^{3}

$15 a b d^{3}$

Step 1

Since

20 c d, 40 a^{2} c^{2} d^{4}, 15 a b d^{3}

$20 c d, 40 a^{2} c^{2} d^{4}, 15 a b d^{3}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part

20, 40, 15

$20, 40, 15$ then find LCM for the variable part

c^{1}, d^{1}, a^{2}, c^{2}, d^{4}, a^{1}, b^{1}, d^{3}

$c^{1}, d^{1}, a^{2}, c^{2}, d^{4}, a^{1}, b^{1}, d^{3}$ .

Step 2

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 3

The prime factors for

20

$20$ are

2 \cdot 2 \cdot 5

$2 \cdot 2 \cdot 5$ .

Tap for more steps...

Step 3.1

20

$20$ has factors of

2

$2$ and

10

$10$ .

2 \cdot 10

$2 \cdot 10$

Step 3.2

10

$10$ has factors of

2

$2$ and

5

$5$ .

2 \cdot 2 \cdot 5

$2 \cdot 2 \cdot 5$

2 \cdot 2 \cdot 5

$2 \cdot 2 \cdot 5$

Step 4

The prime factors for

40

$40$ are

2 \cdot 2 \cdot 2 \cdot 5

$2 \cdot 2 \cdot 2 \cdot 5$ .

Tap for more steps...

Step 4.1

40

$40$ has factors of

2

$2$ and

20

$20$ .

2 \cdot 20

$2 \cdot 20$

Step 4.2

20

$20$ has factors of

2

$2$ and

10

$10$ .

2 \cdot 2 \cdot 10

$2 \cdot 2 \cdot 10$

Step 4.3

10

$10$ has factors of

2

$2$ and

5

$5$ .

2 \cdot 2 \cdot 2 \cdot 5

$2 \cdot 2 \cdot 2 \cdot 5$

2 \cdot 2 \cdot 2 \cdot 5

$2 \cdot 2 \cdot 2 \cdot 5$

Step 5

15

$15$ has factors of

3

$3$ and

5

$5$ .

3 \cdot 5

$3 \cdot 5$

Step 6

The LCM of

20, 40, 15

$20, 40, 15$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

$2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$

Step 7

Multiply

2 \cdot 2 \cdot 2 \cdot 3 \cdot 5

$2 \cdot 2 \cdot 2 \cdot 3 \cdot 5$ .

Tap for more steps...

Step 7.1

Multiply

2

$2$ by

2

$2$ .

4 \cdot 2 \cdot 3 \cdot 5

$4 \cdot 2 \cdot 3 \cdot 5$

Step 7.2

Multiply

4

$4$ by

2

$2$ .

8 \cdot 3 \cdot 5

$8 \cdot 3 \cdot 5$

Step 7.3

Multiply

8

$8$ by

3

$3$ .

24 \cdot 5

$24 \cdot 5$

Step 7.4

Multiply

24

$24$ by

5

$5$ .

120

$120$

120

$120$

Step 8

The factor for

c^{1}

$c^{1}$ is

c

$c$ itself.

c^{1} = c

$c^{1} = c$

c

$c$ occurs

1

$1$ time.

Step 9

The factor for

d^{1}

$d^{1}$ is

d

$d$ itself.

d^{1} = d

$d^{1} = d$

d

$d$ occurs

1

$1$ time.

Step 10

The factors for

a^{2}

$a^{2}$ are

a \cdot a

$a \cdot a$ , which is

a

$a$ multiplied by each other

2

$2$ times.

a^{2} = a \cdot a

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

a

$a$ occurs

2

$2$ times.

Step 11

The factors for

c^{2}

$c^{2}$ are

c \cdot c

$c \cdot c$ , which is

c

$c$ multiplied by each other

2

$2$ times.

c^{2} = c \cdot c

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

c

$c$ occurs

2

$2$ times.

Step 12

The factors for

d^{4}

$d^{4}$ are

d \cdot d \cdot d \cdot d

$d \cdot d \cdot d \cdot d$ , which is

d

$d$ multiplied by each other

4

$4$ times.

d^{4} = d \cdot d \cdot d \cdot d

$d^{4} = d \cdot d \cdot d \cdot d$

d

$d$ occurs

4

$4$ times.

Step 13

The factor for

a^{1}

$a^{1}$ is

a

$a$ itself.

a^{1} = a

$a^{1} = a$

a

$a$ occurs

1

$1$ time.

Step 14

The factor for

b^{1}

$b^{1}$ is

b

$b$ itself.

b^{1} = b

$b^{1} = b$

b

$b$ occurs

1

$1$ time.

Step 15

The factors for

d^{3}

$d^{3}$ are

d \cdot d \cdot d

$d \cdot d \cdot d$ , which is

d

$d$ multiplied by each other

3

$3$ times.

d^{3} = d \cdot d \cdot d

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

d

$d$ occurs

3

$3$ times.

Step 16

The LCM of

c^{1}, d^{1}, a^{2}, c^{2}, d^{4}, a^{1}, b^{1}, d^{3}

$c^{1}, d^{1}, a^{2}, c^{2}, d^{4}, a^{1}, b^{1}, d^{3}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

c \cdot c \cdot d \cdot d \cdot d \cdot d \cdot a \cdot a \cdot b

$c \cdot c \cdot d \cdot d \cdot d \cdot d \cdot a \cdot a \cdot b$

Step 17

Simplify

c \cdot c \cdot d \cdot d \cdot d \cdot d \cdot a \cdot a \cdot b

$c \cdot c \cdot d \cdot d \cdot d \cdot d \cdot a \cdot a \cdot b$ .

Tap for more steps...

Step 17.1

Multiply

c

$c$ by

c

$c$ .

c^{2} \cdot d \cdot d \cdot d \cdot d \cdot a \cdot a \cdot b

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

Step 17.2

Multiply

d

$d$ by

d

$d$ by adding the exponents.

Tap for more steps...

Step 17.2.1

Move

d

$d$ .

c^{2} \cdot (d \cdot d) \cdot d \cdot d \cdot a \cdot a \cdot b

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

Step 17.2.2

Multiply

d

$d$ by

d

$d$ .

c^{2} \cdot d^{2} \cdot d \cdot d \cdot a \cdot a \cdot b

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

c^{2} \cdot d^{2} \cdot d \cdot d \cdot a \cdot a \cdot b

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

Step 17.3

Multiply

d^{2}

$d^{2}$ by

d

$d$ by adding the exponents.

Tap for more steps...

Step 17.3.1

Move

d

$d$ .

c^{2} \cdot (d \cdot d^{2}) \cdot d \cdot a \cdot a \cdot b

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

Step 17.3.2

Multiply

d

$d$ by

d^{2}

$d^{2}$ .

Tap for more steps...

Step 17.3.2.1

Raise

d

$d$ to the power of

1

$1$ .

c^{2} \cdot (d^{1} d^{2}) \cdot d \cdot a \cdot a \cdot b

$c^{2} \cdot (d^{1} d^{2}) \cdot d \cdot a \cdot a \cdot b$

Step 17.3.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

c^{2} \cdot d^{1 + 2} \cdot d \cdot a \cdot a \cdot b

$c^{2} \cdot d^{1 + 2} \cdot d \cdot a \cdot a \cdot b$

c^{2} \cdot d^{1 + 2} \cdot d \cdot a \cdot a \cdot b

$c^{2} \cdot d^{1 + 2} \cdot d \cdot a \cdot a \cdot b$

Step 17.3.3

Add

1

$1$ and

2

$2$ .

c^{2} \cdot d^{3} \cdot d \cdot a \cdot a \cdot b

$c^{2} \cdot d^{3} \cdot d \cdot a \cdot a \cdot b$

c^{2} \cdot d^{3} \cdot d \cdot a \cdot a \cdot b

$c^{2} \cdot d^{3} \cdot d \cdot a \cdot a \cdot b$

Step 17.4

Multiply

d^{3}

$d^{3}$ by

d

$d$ by adding the exponents.

Tap for more steps...

Step 17.4.1

Move

d

$d$ .

c^{2} \cdot (d \cdot d^{3}) \cdot a \cdot a \cdot b

$c^{2} \cdot (d \cdot d^{3}) \cdot a \cdot a \cdot b$

Step 17.4.2

Multiply

d

$d$ by

d^{3}

$d^{3}$ .

Tap for more steps...

Step 17.4.2.1

Raise

d

$d$ to the power of

1

$1$ .

c^{2} \cdot (d^{1} d^{3}) \cdot a \cdot a \cdot b

$c^{2} \cdot (d^{1} d^{3}) \cdot a \cdot a \cdot b$

Step 17.4.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

c^{2} \cdot d^{1 + 3} \cdot a \cdot a \cdot b

$c^{2} \cdot d^{1 + 3} \cdot a \cdot a \cdot b$

c^{2} \cdot d^{1 + 3} \cdot a \cdot a \cdot b

$c^{2} \cdot d^{1 + 3} \cdot a \cdot a \cdot b$

Step 17.4.3

Add

1

$1$ and

3

$3$ .

c^{2} \cdot d^{4} \cdot a \cdot a \cdot b

$c^{2} \cdot d^{4} \cdot a \cdot a \cdot b$

c^{2} \cdot d^{4} \cdot a \cdot a \cdot b

$c^{2} \cdot d^{4} \cdot a \cdot a \cdot b$

Step 17.5

Multiply

a

$a$ by

a

$a$ by adding the exponents.

Tap for more steps...

Step 17.5.1

Move

a

$a$ .

c^{2} \cdot d^{4} \cdot (a \cdot a) \cdot b

$c^{2} \cdot d^{4} \cdot (a \cdot a) \cdot b$

Step 17.5.2

Multiply

a

$a$ by

a

$a$ .

c^{2} \cdot d^{4} \cdot a^{2} \cdot b

$c^{2} \cdot d^{4} \cdot a^{2} \cdot b$

c^{2} d^{4} a^{2} b

$c^{2} d^{4} a^{2} b$

c^{2} d^{4} a^{2} b

$c^{2} d^{4} a^{2} b$

Step 18

The LCM for

20 c d, 40 a^{2} c^{2} d^{4}, 15 a b d^{3}

$20 c d, 40 a^{2} c^{2} d^{4}, 15 a b d^{3}$ is the numeric part

120

$120$ multiplied by the variable part.

120 c^{2} d^{4} a^{2} b

$120 c^{2} d^{4} a^{2} b$