Answer :
To find the least common multiple (LCM) of the expressions [tex]\(12 v y^3 x^5\)[/tex] and [tex]\(9 y^8 x^7\)[/tex], we will break down each part of the expression separately.
### Step 1: Factorize the Numerical Coefficients
- The number 12 can be factorized as [tex]\(2^2 \times 3\)[/tex].
- The number 9 can be factorized as [tex]\(3^2\)[/tex].
To find the LCM of the numerical parts, take the highest power of each prime number present in the factorizations:
- Highest power of 2: [tex]\(2^2\)[/tex]
- Highest power of 3: [tex]\(3^2\)[/tex]
Therefore, the LCM of 12 and 9 is:
[tex]\[ 2^2 \times 3^2 = 4 \times 9 = 36 \][/tex]
### Step 2: Determine the LCM of Variable Parts
For each variable, we choose the highest exponent between the two expressions.
- Variable [tex]\(v\)[/tex]:
- Only appears in the first expression as [tex]\(v^1\)[/tex].
- Therefore, we take [tex]\(v^1\)[/tex].
- Variable [tex]\(y\)[/tex]:
- Appears as [tex]\(y^3\)[/tex] in the first expression and [tex]\(y^8\)[/tex] in the second.
- The highest power is [tex]\(y^8\)[/tex].
- Variable [tex]\(x\)[/tex]:
- Appears as [tex]\(x^5\)[/tex] in the first expression and [tex]\(x^7\)[/tex] in the second.
- The highest power is [tex]\(x^7\)[/tex].
### Step 3: Combine the Results
To find the LCM of the entire expressions, combine the LCM of the numerical parts with the highest powers of each variable:
[tex]\[ \text{LCM} = 36 v^1 y^8 x^7 \][/tex]
Thus, the least common multiple of the expressions [tex]\(12 v y^3 x^5\)[/tex] and [tex]\(9 y^8 x^7\)[/tex] is:
[tex]\[ 36v y^8 x^7 \][/tex]
### Step 1: Factorize the Numerical Coefficients
- The number 12 can be factorized as [tex]\(2^2 \times 3\)[/tex].
- The number 9 can be factorized as [tex]\(3^2\)[/tex].
To find the LCM of the numerical parts, take the highest power of each prime number present in the factorizations:
- Highest power of 2: [tex]\(2^2\)[/tex]
- Highest power of 3: [tex]\(3^2\)[/tex]
Therefore, the LCM of 12 and 9 is:
[tex]\[ 2^2 \times 3^2 = 4 \times 9 = 36 \][/tex]
### Step 2: Determine the LCM of Variable Parts
For each variable, we choose the highest exponent between the two expressions.
- Variable [tex]\(v\)[/tex]:
- Only appears in the first expression as [tex]\(v^1\)[/tex].
- Therefore, we take [tex]\(v^1\)[/tex].
- Variable [tex]\(y\)[/tex]:
- Appears as [tex]\(y^3\)[/tex] in the first expression and [tex]\(y^8\)[/tex] in the second.
- The highest power is [tex]\(y^8\)[/tex].
- Variable [tex]\(x\)[/tex]:
- Appears as [tex]\(x^5\)[/tex] in the first expression and [tex]\(x^7\)[/tex] in the second.
- The highest power is [tex]\(x^7\)[/tex].
### Step 3: Combine the Results
To find the LCM of the entire expressions, combine the LCM of the numerical parts with the highest powers of each variable:
[tex]\[ \text{LCM} = 36 v^1 y^8 x^7 \][/tex]
Thus, the least common multiple of the expressions [tex]\(12 v y^3 x^5\)[/tex] and [tex]\(9 y^8 x^7\)[/tex] is:
[tex]\[ 36v y^8 x^7 \][/tex]
