Finite Math Examples

$10 m^{3}$ , $4 w^{4}$

Step 1

Since $10 m^{3}, 4 w^{4}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $10, 4$ then find LCM for the variable part $m^{3}, w^{4}$ .

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

$10$ has factors of $2$ and $5$ .

$2 \cdot 5$

Step 4

$4$ has factors of $2$ and $2$ .

$2 \cdot 2$

Step 5

The LCM of $10, 4$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 5$

Step 6

Multiply $2 \cdot 2 \cdot 5$ .

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Step 6.1

Multiply $2$ by $2$ .

$4 \cdot 5$

Step 6.2

Multiply $4$ by $5$ .

$20$

Step 7

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 8

The factors for $w^{4}$ are $w \cdot w \cdot w \cdot w$ , which is $w$ multiplied by each other $4$ times.

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

$w$ occurs $4$ times.

Step 9

The LCM of $m^{3}, w^{4}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$m \cdot m \cdot m \cdot w \cdot w \cdot w \cdot w$

Step 10

Simplify $m \cdot m \cdot m \cdot w \cdot w \cdot w \cdot w$ .

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Step 10.1

Multiply $m$ by $m$ .

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

Step 10.2

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

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Step 10.2.1

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

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Step 10.2.1.1

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

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

Step 10.2.1.2

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

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

Step 10.2.2

Add $2$ and $1$ .

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

Step 10.3

Multiply $w$ by $w$ by adding the exponents.

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Step 10.3.1

Move $w$ .

$m^{3} \cdot (w \cdot w) \cdot w \cdot w$

Step 10.3.2

Multiply $w$ by $w$ .

$m^{3} \cdot w^{2} \cdot w \cdot w$

Step 10.4

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

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Step 10.4.1

Move $w$ .

$m^{3} \cdot (w \cdot w^{2}) \cdot w$

Step 10.4.2

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

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Step 10.4.2.1

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

$m^{3} \cdot (w^{1} w^{2}) \cdot w$

Step 10.4.2.2

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

$m^{3} \cdot w^{1 + 2} \cdot w$

Step 10.4.3

Add $1$ and $2$ .

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

Step 10.5

Multiply $w^{3}$ by $w$ by adding the exponents.

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Step 10.5.1

Move $w$ .

$m^{3} \cdot (w \cdot w^{3})$

Step 10.5.2

Multiply $w$ by $w^{3}$ .

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Step 10.5.2.1

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

$m^{3} \cdot (w^{1} w^{3})$

Step 10.5.2.2

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

$m^{3} \cdot w^{1 + 3}$

Step 10.5.3

Add $1$ and $3$ .

$m^{3} \cdot w^{4}$

$m^{3} w^{4}$

Step 11

The LCM for $10 m^{3}, 4 w^{4}$ is the numeric part $20$ multiplied by the variable part.

$20 m^{3} w^{4}$