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To find the least common multiple (LCM) of the two expressions [tex]\( 9w^8v^6x^4 \)[/tex] and [tex]\( 6v^7x^5 \)[/tex], we need to determine the highest power of each factor (both numerical and variable) that appears in either expression. Let's decompose each expression into its prime factors and variables:
### Step-by-Step Decomposition:
1. Decomposing [tex]\( 9w^8v^6x^4 \)[/tex]:
- [tex]\( 9 \)[/tex] can be written as [tex]\( 3^2 \)[/tex].
- Therefore, the expression becomes [tex]\( 3^2 w^8 v^6 x^4 \)[/tex].
2. Decomposing [tex]\( 6v^7x^5 \)[/tex]:
- [tex]\( 6 \)[/tex] can be written as [tex]\( 2 \times 3 \)[/tex].
- Therefore, the expression becomes [tex]\( 2 \times 3 v^7 x^5 \)[/tex].
### Building the LCM:
The LCM will take the highest power of each factor appearing in both expressions. We need to consider the following factors: [tex]\( 2 \)[/tex], [tex]\( 3 \)[/tex], [tex]\( w \)[/tex], [tex]\( v \)[/tex], and [tex]\( x \)[/tex].
1. Highest Power of 2:
- The first expression [tex]\( 3^2w^8v^6x^4 \)[/tex] does not have any factor of 2.
- The second expression [tex]\( 2 \times 3 v^7 x^5 \)[/tex] has [tex]\( 2^1 \)[/tex].
- Therefore, we take [tex]\( 2^1 \)[/tex].
2. Highest Power of 3:
- The first expression [tex]\( 3^2 w^8 v^6 x^4 \)[/tex] has [tex]\( 3^2 \)[/tex].
- The second expression [tex]\( 2 \times 3 v^7 x^5 \)[/tex] has [tex]\( 3^1 \)[/tex].
- Therefore, we take [tex]\( 3^2 \)[/tex].
3. Highest Power of [tex]\( w \)[/tex]:
- The first expression [tex]\( 3^2 w^8 v^6 x^4 \)[/tex] has [tex]\( w^8 \)[/tex].
- The second expression [tex]\( 2 \times 3 v^7 x^5 \)[/tex] does not have [tex]\( w \)[/tex].
- Therefore, we take [tex]\( w^8 \)[/tex].
4. Highest Power of [tex]\( v \)[/tex]:
- The first expression [tex]\( 3^2 w^8 v^6 x^4 \)[/tex] has [tex]\( v^6 \)[/tex].
- The second expression [tex]\( 2 \times 3 v^7 x^5 \)[/tex] has [tex]\( v^7 \)[/tex].
- Therefore, we take [tex]\( v^7 \)[/tex].
5. Highest Power of [tex]\( x \)[/tex]:
- The first expression [tex]\( 3^2 w^8 v^6 x^4 \)[/tex] has [tex]\( x^4 \)[/tex].
- The second expression [tex]\( 2 \times 3 v^7 x^5 \)[/tex] has [tex]\( x^5 \)[/tex].
- Therefore, we take [tex]\( x^5 \)[/tex].
### Composing the LCM:
Putting it all together, the LCM of the two expressions will be:
[tex]\[ \boxed{2 \times 3^2 \times w^8 \times v^7 \times x^5} \][/tex]
Simplifying further:
[tex]\[ \boxed{18 w^8 v^7 x^5} \][/tex]
So, the LCM of [tex]\( 9w^8v^6x^4 \)[/tex] and [tex]\( 6v^7x^5 \)[/tex] is [tex]\( \boxed{18 w^8 v^7 x^5} \)[/tex].
### Step-by-Step Decomposition:
1. Decomposing [tex]\( 9w^8v^6x^4 \)[/tex]:
- [tex]\( 9 \)[/tex] can be written as [tex]\( 3^2 \)[/tex].
- Therefore, the expression becomes [tex]\( 3^2 w^8 v^6 x^4 \)[/tex].
2. Decomposing [tex]\( 6v^7x^5 \)[/tex]:
- [tex]\( 6 \)[/tex] can be written as [tex]\( 2 \times 3 \)[/tex].
- Therefore, the expression becomes [tex]\( 2 \times 3 v^7 x^5 \)[/tex].
### Building the LCM:
The LCM will take the highest power of each factor appearing in both expressions. We need to consider the following factors: [tex]\( 2 \)[/tex], [tex]\( 3 \)[/tex], [tex]\( w \)[/tex], [tex]\( v \)[/tex], and [tex]\( x \)[/tex].
1. Highest Power of 2:
- The first expression [tex]\( 3^2w^8v^6x^4 \)[/tex] does not have any factor of 2.
- The second expression [tex]\( 2 \times 3 v^7 x^5 \)[/tex] has [tex]\( 2^1 \)[/tex].
- Therefore, we take [tex]\( 2^1 \)[/tex].
2. Highest Power of 3:
- The first expression [tex]\( 3^2 w^8 v^6 x^4 \)[/tex] has [tex]\( 3^2 \)[/tex].
- The second expression [tex]\( 2 \times 3 v^7 x^5 \)[/tex] has [tex]\( 3^1 \)[/tex].
- Therefore, we take [tex]\( 3^2 \)[/tex].
3. Highest Power of [tex]\( w \)[/tex]:
- The first expression [tex]\( 3^2 w^8 v^6 x^4 \)[/tex] has [tex]\( w^8 \)[/tex].
- The second expression [tex]\( 2 \times 3 v^7 x^5 \)[/tex] does not have [tex]\( w \)[/tex].
- Therefore, we take [tex]\( w^8 \)[/tex].
4. Highest Power of [tex]\( v \)[/tex]:
- The first expression [tex]\( 3^2 w^8 v^6 x^4 \)[/tex] has [tex]\( v^6 \)[/tex].
- The second expression [tex]\( 2 \times 3 v^7 x^5 \)[/tex] has [tex]\( v^7 \)[/tex].
- Therefore, we take [tex]\( v^7 \)[/tex].
5. Highest Power of [tex]\( x \)[/tex]:
- The first expression [tex]\( 3^2 w^8 v^6 x^4 \)[/tex] has [tex]\( x^4 \)[/tex].
- The second expression [tex]\( 2 \times 3 v^7 x^5 \)[/tex] has [tex]\( x^5 \)[/tex].
- Therefore, we take [tex]\( x^5 \)[/tex].
### Composing the LCM:
Putting it all together, the LCM of the two expressions will be:
[tex]\[ \boxed{2 \times 3^2 \times w^8 \times v^7 \times x^5} \][/tex]
Simplifying further:
[tex]\[ \boxed{18 w^8 v^7 x^5} \][/tex]
So, the LCM of [tex]\( 9w^8v^6x^4 \)[/tex] and [tex]\( 6v^7x^5 \)[/tex] is [tex]\( \boxed{18 w^8 v^7 x^5} \)[/tex].
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