Precalculus Examples

$\frac{7}{6 x^{2}}$ , $\frac{9}{14 x^{3}}$

Step 1

Since $\frac{7}{6 x^{2}}, \frac{9}{14 x^{3}}$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $\frac{7}{6}, \frac{9}{14}$ then find LCM for the variable part $x^{2}, x^{3}$ .

Step 2

To find the LCM for a list of fractions, check if denominators are similar or not.

Fractions with the same denominator:

1: $LCM (\frac{a}{b}, \frac{c}{b}) = \frac{LCM (a, c)}{b}$

Fractions with different denominators such as, $LCM (\frac{a}{b}, \frac{c}{d})$ :

1: Find the LCM of $b$ and $d = LCM (b, d)$

2: Multiply the numerator and denominator of the first fraction $\frac{a}{b}$ by $\frac{LCM (b, d)}{b}$

3: Multiply the numerator and denominator of the second fraction $\frac{c}{d}$ by $\frac{LCM (b, d)}{d}$

4: After making the denominators for all the fractions same, in this case, only two fractions, find the LCM of the new numerators

5: The LCM will be the $\frac{LCM (numerators)}{LCM (b, d)}$

Step 3

Find the LCM for the denominators of $\frac{7}{6 x^{2}}, \frac{9}{14 x^{3}}$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

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

$N \cdot N a, N a N$

Step 3.1.2

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

$N^{1 + 1} a, N a N$

Step 3.1.3

Add $1$ and $1$ .

$N^{2} a, N a N$

Step 3.1.4

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

$N^{2} a, N \cdot N a$

Step 3.1.5

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

$N^{2} a, N^{1 + 1} a$

Step 3.1.6

Add $1$ and $1$ .

$N^{2} a, N^{2} a$

Step 3.2

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

Step 3.3

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 3.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 3.5

The LCM of $1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 3.6

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

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

$N$ occurs $2$ times.

Step 3.7

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

$a^{1} = a$

$a$ occurs $1$ time.

Step 3.8

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

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

$N$ occurs $2$ times.

Step 3.9

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

$a^{1} = a$

$a$ occurs $1$ time.

Step 3.10

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

$N \cdot N \cdot a$

Step 3.11

Multiply $N$ by $N$ .

$N^{2} a$

Step 4

Multiply each number by $\frac{n}{n}$ , where $n$ is a number that makes the denominator $N^{2} a$ .

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Step 4.1

Multiply the numerator and denominator of $\frac{7}{6 x^{2}}$ by $\frac{N^{2} a}{6 x^{2}}$ .

$\frac{7 \cdot \frac{N^{2} a}{6 x^{2}}}{6 x^{2} \cdot \frac{N^{2} a}{6 x^{2}}}$

Step 4.2

Combine $7$ and $\frac{N^{2} a}{6 x^{2}}$ .

$\frac{\frac{7 N^{2} a}{6 x^{2}}}{6 x^{2} \cdot \frac{N^{2} a}{6 x^{2}}}$

Step 4.3

Cancel the common factor of $6 x^{2}$ .

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Step 4.3.1

Cancel the common factor.

$\frac{7 N^{2} a}{6 x^{2}} \cdot \frac{N^{2} a}{6 x^{2}}$

Step 4.3.2

Rewrite the expression.

$\frac{7 N^{2} a}{N^{2} a}$

Step 4.4

Multiply the numerator and denominator of $\frac{9}{14 x^{3}}$ by $\frac{7 N^{2} a}{N^{2} a}$ .

$\frac{9 \cdot \frac{7 N^{2} a}{N^{2} a}}{14 x^{3} \cdot \frac{7 N^{2} a}{N^{2} a}}$

Step 4.5

Cancel the common factor of $N^{2}$ .

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Step 4.5.1

Cancel the common factor.

$9 \cdot \frac{\frac{7 N^{2} a}{N^{2} a}}{14 x^{3} \cdot \frac{7 N^{2} a}{N^{2} a}}$

Step 4.5.2

Rewrite the expression.

$9 \cdot \frac{\frac{7 a}{a}}{14 x^{3} \cdot \frac{7 N^{2} a}{N^{2} a}}$

Step 4.6

Cancel the common factor of $a$ .

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Step 4.6.1

Cancel the common factor.

$9 \cdot \frac{\frac{7 a}{a}}{14 x^{3} \cdot \frac{7 N^{2} a}{N^{2} a}}$

Step 4.6.2

Divide $7$ by $1$ .

$9 \cdot \frac{7}{14 x^{3} \cdot \frac{7 N^{2} a}{N^{2} a}}$

Step 4.7

Multiply $9$ by $7$ .

$\frac{63}{14 x^{3} \cdot \frac{7 N^{2} a}{N^{2} a}}$

Step 4.8

Cancel the common factor of $N^{2}$ .

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Step 4.8.1

Cancel the common factor.

$\frac{63}{14 x^{3}} \cdot \frac{7 N^{2} a}{N^{2} a}$

Step 4.8.2

Rewrite the expression.

$\frac{63}{14 x^{3}} \cdot \frac{7 a}{a}$

Step 4.9

Cancel the common factor of $a$ .

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Step 4.9.1

Cancel the common factor.

$\frac{63}{14 x^{3}} \cdot \frac{7 a}{a}$

Step 4.9.2

Divide $7$ by $1$ .

$\frac{63}{14 x^{3}} \cdot 7$

Step 4.10

Multiply $7$ by $14$ .

$\frac{63}{98 x^{3}}$

Step 4.11

Write the new list with the same denominators.

$\frac{7 N^{2} a}{N^{2} a}, \frac{63}{98 x^{3}}$

Step 5

Find the LCM for $7 N^{2} a, 63$ .

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Step 5.1

Since $7 N^{2} a, 63$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $7, 63$ then find LCM for the variable part $N^{2}, a^{1}$ .

Step 5.2

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 5.3

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

$7$ is a prime number

Step 5.4

The prime factors for $63$ are $3 \cdot 3 \cdot 7$ .

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Step 5.4.1

$63$ has factors of $3$ and $21$ .

$3 \cdot 21$

Step 5.4.2

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

$3 \cdot 3 \cdot 7$

Step 5.5

Multiply $3 \cdot 3 \cdot 7$ .

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Step 5.5.1

Multiply $3$ by $3$ .

$9 \cdot 7$

Step 5.5.2

Multiply $9$ by $7$ .

$63$

Step 5.6

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

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

$N$ occurs $2$ times.

Step 5.7

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

$a^{1} = a$

$a$ occurs $1$ time.

Step 5.8

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

$N \cdot N \cdot a$

Step 5.9

Multiply $N$ by $N$ .

$N^{2} a$

Step 5.10

The LCM for $7 N^{2} a, 63$ is the numeric part $63$ multiplied by the variable part.

$63 N^{2} a$

Step 6

The answer can be found by taking the LCM of $7 N^{2} a, 63$ and dividing by the LCM of $6, 14$ .

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Step 6.1

Divide the LCM of $7 N^{2} a, 63$ by the LCM of $6, 14$ .

$\frac{63 N^{2} a}{N^{2} a}$

Step 6.2

Cancel the common factor of $N^{2}$ .

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Step 6.2.1

Cancel the common factor.

$\frac{63 N^{2} a}{N^{2} a}$

Step 6.2.2

Rewrite the expression.

$\frac{63 a}{a}$

Step 6.3

Cancel the common factor of $a$ .

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Step 6.3.1

Cancel the common factor.

$\frac{63 a}{a}$

Step 6.3.2

Divide $63$ by $1$ .

$63$

Step 7

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

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

$x$ occurs $2$ times.

Step 8

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

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

$x$ occurs $3$ times.

Step 9

The LCM of $x^{2}, x^{3}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x \cdot x$

Step 10

Simplify $x \cdot x \cdot x$ .

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Step 10.1

Multiply $x$ by $x$ .

$x^{2} \cdot x$

Step 10.2

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

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Step 10.2.1

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

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Step 10.2.1.1

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

$x^{2} \cdot x^{1}$

Step 10.2.1.2

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

$x^{2 + 1}$

Step 10.2.2

Add $2$ and $1$ .

$x^{3}$

Step 11

The LCM for $\frac{7}{6 x^{2}}, \frac{9}{14 x^{3}}$ is the numeric part $63$ multiplied by the variable part.

$63 x^{3}$