Answer :
To find the least common multiple (LCM) of two algebraic expressions, we need to consider the numerical coefficients and the variables separately. We handle each component by examining its highest power in the expressions.
Let's break down the expressions:
1. Expression 1: [tex]\( 12y^8v^2x^7 \)[/tex]
- Coefficient: 12
- Variables: [tex]\( y^8 \)[/tex], [tex]\( v^2 \)[/tex], [tex]\( x^7 \)[/tex]
2. Expression 2: [tex]\( 9y^4v^5 \)[/tex]
- Coefficient: 9
- Variables: [tex]\( y^4 \)[/tex], [tex]\( v^5 \)[/tex]
Step 1: Find the LCM of the numerical coefficients.
- The coefficients are 12 and 9.
- To find the LCM of 12 and 9, list their prime factors:
- [tex]\( 12 = 2^2 \times 3 \)[/tex]
- [tex]\( 9 = 3^2 \)[/tex]
- The LCM is found by taking the highest power of each prime:
- Highest power of 2: [tex]\( 2^2 \)[/tex]
- Highest power of 3: [tex]\( 3^2 \)[/tex]
- LCM of coefficients = [tex]\( 2^2 \times 3^2 = 4 \times 9 = 36 \)[/tex]
Step 2: Determine the LCM for each variable with its highest exponent.
- Variable [tex]\( y \)[/tex]:
- Expression 1: [tex]\( y^8 \)[/tex]
- Expression 2: [tex]\( y^4 \)[/tex]
- Highest power of [tex]\( y \)[/tex]: [tex]\( y^8 \)[/tex]
- Variable [tex]\( v \)[/tex]:
- Expression 1: [tex]\( v^2 \)[/tex]
- Expression 2: [tex]\( v^5 \)[/tex]
- Highest power of [tex]\( v \)[/tex]: [tex]\( v^5 \)[/tex]
- Variable [tex]\( x \)[/tex]:
- Only present in Expression 1: [tex]\( x^7 \)[/tex]
- Highest power of [tex]\( x \)[/tex]: [tex]\( x^7 \)[/tex]
Step 3: Combine the LCM of the coefficients and the highest powers of the variables.
The least common multiple of the expressions is:
[tex]\[ \text{LCM} = 36y^8v^5x^7 \][/tex]
So, the LCM of [tex]\( 12y^8v^2x^7 \)[/tex] and [tex]\( 9y^4v^5 \)[/tex] is [tex]\( 36y^8v^5x^7 \)[/tex].
Let's break down the expressions:
1. Expression 1: [tex]\( 12y^8v^2x^7 \)[/tex]
- Coefficient: 12
- Variables: [tex]\( y^8 \)[/tex], [tex]\( v^2 \)[/tex], [tex]\( x^7 \)[/tex]
2. Expression 2: [tex]\( 9y^4v^5 \)[/tex]
- Coefficient: 9
- Variables: [tex]\( y^4 \)[/tex], [tex]\( v^5 \)[/tex]
Step 1: Find the LCM of the numerical coefficients.
- The coefficients are 12 and 9.
- To find the LCM of 12 and 9, list their prime factors:
- [tex]\( 12 = 2^2 \times 3 \)[/tex]
- [tex]\( 9 = 3^2 \)[/tex]
- The LCM is found by taking the highest power of each prime:
- Highest power of 2: [tex]\( 2^2 \)[/tex]
- Highest power of 3: [tex]\( 3^2 \)[/tex]
- LCM of coefficients = [tex]\( 2^2 \times 3^2 = 4 \times 9 = 36 \)[/tex]
Step 2: Determine the LCM for each variable with its highest exponent.
- Variable [tex]\( y \)[/tex]:
- Expression 1: [tex]\( y^8 \)[/tex]
- Expression 2: [tex]\( y^4 \)[/tex]
- Highest power of [tex]\( y \)[/tex]: [tex]\( y^8 \)[/tex]
- Variable [tex]\( v \)[/tex]:
- Expression 1: [tex]\( v^2 \)[/tex]
- Expression 2: [tex]\( v^5 \)[/tex]
- Highest power of [tex]\( v \)[/tex]: [tex]\( v^5 \)[/tex]
- Variable [tex]\( x \)[/tex]:
- Only present in Expression 1: [tex]\( x^7 \)[/tex]
- Highest power of [tex]\( x \)[/tex]: [tex]\( x^7 \)[/tex]
Step 3: Combine the LCM of the coefficients and the highest powers of the variables.
The least common multiple of the expressions is:
[tex]\[ \text{LCM} = 36y^8v^5x^7 \][/tex]
So, the LCM of [tex]\( 12y^8v^2x^7 \)[/tex] and [tex]\( 9y^4v^5 \)[/tex] is [tex]\( 36y^8v^5x^7 \)[/tex].
