Answer :
To find the least common multiple (LCM) of the expressions [tex]\(6 y^4 u^3\)[/tex] and [tex]\(9 y^4 u^8 x^7\)[/tex], we need to follow these steps:
1. Identify the coefficients and variables involved:
- The first expression is [tex]\(6 y^4 u^3\)[/tex]. It has the coefficient 6 and the variables [tex]\(y^4\)[/tex] and [tex]\(u^3\)[/tex].
- The second expression is [tex]\(9 y^4 u^8 x^7\)[/tex]. It has the coefficient 9 and the variables [tex]\(y^4\)[/tex], [tex]\(u^8\)[/tex], and [tex]\(x^7\)[/tex].
2. Find the LCM of the coefficients:
- The coefficients are 6 and 9.
- The prime factorization of 6 is [tex]\(2 \times 3\)[/tex].
- The prime factorization of 9 is [tex]\(3^2\)[/tex].
- The LCM of 6 and 9 is obtained by taking the highest power of each prime factor. Thus, the LCM of 6 and 9 is [tex]\(2 \times 3^2 = 18\)[/tex].
3. Find the LCM of each variable with respect to its powers:
- For [tex]\(y\)[/tex]:
- The first expression has [tex]\(y^4\)[/tex].
- The second expression also has [tex]\(y^4\)[/tex].
- The LCM of [tex]\(y^4\)[/tex] and [tex]\(y^4\)[/tex] is [tex]\(y^4\)[/tex].
- For [tex]\(u\)[/tex]:
- The first expression has [tex]\(u^3\)[/tex].
- The second expression has [tex]\(u^8\)[/tex].
- The LCM of [tex]\(u^3\)[/tex] and [tex]\(u^8\)[/tex] is [tex]\(u^8\)[/tex] (since [tex]\(u^8\)[/tex] is the highest power).
- For [tex]\(x\)[/tex]:
- The first expression does not have [tex]\(x\)[/tex].
- The second expression has [tex]\(x^7\)[/tex].
- The LCM takes the highest power available, which is [tex]\(x^7\)[/tex].
4. Combine all parts to form the LCM of the entire expressions:
- The LCM of the coefficients is 18.
- The LCM of the variable parts [tex]\(y\)[/tex], [tex]\(u\)[/tex], and [tex]\(x\)[/tex] are [tex]\(y^4\)[/tex], [tex]\(u^8\)[/tex], and [tex]\(x^7\)[/tex] respectively.
Therefore, the least common multiple (LCM) of [tex]\(6 y^4 u^3\)[/tex] and [tex]\(9 y^4 u^8 x^7\)[/tex] is:
[tex]\[18 u^8 x^7 y^4\][/tex]
This is the required least common multiple of the given expressions.
1. Identify the coefficients and variables involved:
- The first expression is [tex]\(6 y^4 u^3\)[/tex]. It has the coefficient 6 and the variables [tex]\(y^4\)[/tex] and [tex]\(u^3\)[/tex].
- The second expression is [tex]\(9 y^4 u^8 x^7\)[/tex]. It has the coefficient 9 and the variables [tex]\(y^4\)[/tex], [tex]\(u^8\)[/tex], and [tex]\(x^7\)[/tex].
2. Find the LCM of the coefficients:
- The coefficients are 6 and 9.
- The prime factorization of 6 is [tex]\(2 \times 3\)[/tex].
- The prime factorization of 9 is [tex]\(3^2\)[/tex].
- The LCM of 6 and 9 is obtained by taking the highest power of each prime factor. Thus, the LCM of 6 and 9 is [tex]\(2 \times 3^2 = 18\)[/tex].
3. Find the LCM of each variable with respect to its powers:
- For [tex]\(y\)[/tex]:
- The first expression has [tex]\(y^4\)[/tex].
- The second expression also has [tex]\(y^4\)[/tex].
- The LCM of [tex]\(y^4\)[/tex] and [tex]\(y^4\)[/tex] is [tex]\(y^4\)[/tex].
- For [tex]\(u\)[/tex]:
- The first expression has [tex]\(u^3\)[/tex].
- The second expression has [tex]\(u^8\)[/tex].
- The LCM of [tex]\(u^3\)[/tex] and [tex]\(u^8\)[/tex] is [tex]\(u^8\)[/tex] (since [tex]\(u^8\)[/tex] is the highest power).
- For [tex]\(x\)[/tex]:
- The first expression does not have [tex]\(x\)[/tex].
- The second expression has [tex]\(x^7\)[/tex].
- The LCM takes the highest power available, which is [tex]\(x^7\)[/tex].
4. Combine all parts to form the LCM of the entire expressions:
- The LCM of the coefficients is 18.
- The LCM of the variable parts [tex]\(y\)[/tex], [tex]\(u\)[/tex], and [tex]\(x\)[/tex] are [tex]\(y^4\)[/tex], [tex]\(u^8\)[/tex], and [tex]\(x^7\)[/tex] respectively.
Therefore, the least common multiple (LCM) of [tex]\(6 y^4 u^3\)[/tex] and [tex]\(9 y^4 u^8 x^7\)[/tex] is:
[tex]\[18 u^8 x^7 y^4\][/tex]
This is the required least common multiple of the given expressions.
