To find the least common multiple (LCM) of the expressions [tex]\( 8 x^3 u^6 \)[/tex] and [tex]\( 12 x^7 u^2 v^5 \)[/tex], we need to determine the LCM of the coefficients and the highest powers of each variable that appear in the expressions. Let's break this down step-by-step:
1. Identify the coefficients and the variables with their exponents:
- First expression: [tex]\( 8 x^3 u^6 \)[/tex]
- Coefficient: 8
- Variables: [tex]\( x^3 \)[/tex], [tex]\( u^6 \)[/tex]
- Second expression: [tex]\( 12 x^7 u^2 v^5 \)[/tex]
- Coefficient: 12
- Variables: [tex]\( x^7 \)[/tex], [tex]\( u^2 \)[/tex], [tex]\( v^5 \)[/tex]
2. Find the LCM of the coefficients:
- Coefficients are 8 and 12.
- Factorize the coefficients:
- [tex]\( 8 = 2^3 \)[/tex]
- [tex]\( 12 = 2^2 \times 3 \)[/tex]
- The LCM of the coefficients is obtained by taking the highest power of each prime factor:
- [tex]\( \text{LCM}(8, 12) = 2^3 \times 3 = 24 \)[/tex]
3. Determine the highest powers of each variable:
- For [tex]\( x \)[/tex]: [tex]\( x^3 \)[/tex] and [tex]\( x^7 \)[/tex]. The highest power is [tex]\( x^7 \)[/tex].
- For [tex]\( u \)[/tex]: [tex]\( u^6 \)[/tex] and [tex]\( u^2 \)[/tex]. The highest power is [tex]\( u^6 \)[/tex].
- For [tex]\( v \)[/tex]: [tex]\( v \)[/tex] appears only in the second expression, where it is [tex]\( v^5 \)[/tex]. The highest power is [tex]\( v^5 \)[/tex].
4. Construct the LCM using the coefficients and the highest powers of each variable:
- Coefficient: 24
- Variables: [tex]\( x^7 \)[/tex], [tex]\( u^6 \)[/tex], [tex]\( v^5 \)[/tex]
Putting it all together, the LCM of the two expressions is:
[tex]\[
\boxed{24 x^7 u^6 v^5}
\][/tex]
So, the least common multiple of [tex]\( 8 x^3 u^6 \)[/tex] and [tex]\( 12 x^7 u^2 v^5 \)[/tex] is [tex]\( 24 x^7 u^6 v^5 \)[/tex].
