To find the least common multiple (LCM) of the two monomials [tex]\( 24x^4w^6y \)[/tex] and [tex]\( 12x^7w^3 \)[/tex], we need to follow these steps:
### Step 1: Find the LCM of the coefficients
The coefficients of the given monomials are 24 and 12. The LCM of 24 and 12 is the smallest number that both 24 and 12 can divide into exactly.
- The prime factorization of 24 is [tex]\( 2^3 \times 3 \)[/tex].
- The prime factorization of 12 is [tex]\( 2^2 \times 3 \)[/tex].
To find the LCM, we take the highest power of each prime factor that appears in the factorizations:
- For the prime [tex]\( 2 \)[/tex], the highest power is [tex]\( 2^3 \)[/tex].
- For the prime [tex]\( 3 \)[/tex], the highest power is [tex]\( 3^1 \)[/tex].
Thus, the LCM of the coefficients is:
[tex]\[ 2^3 \times 3 = 8 \times 3 = 24 \][/tex]
### Step 2: Find the LCM of the variable parts
For the variables [tex]\( x \)[/tex], [tex]\( w \)[/tex], and [tex]\( y \)[/tex]:
#### Variable [tex]\( x \)[/tex]
The exponents of [tex]\( x \)[/tex] are 4 and 7. The LCM for the powers is the highest exponent:
[tex]\[ \text{LCM of } x^4 \text{ and } x^7 \text{ is } x^7 \][/tex]
#### Variable [tex]\( w \)[/tex]
The exponents of [tex]\( w \)[/tex] are 6 and 3. The LCM for the powers is the highest exponent:
[tex]\[ \text{LCM of } w^6 \text{ and } w^3 \text{ is } w^6 \][/tex]
#### Variable [tex]\( y \)[/tex]
[tex]\( y \)[/tex] appears only in the first monomial with a power of 1:
[tex]\[ \text{LCM of } y \text{ and no } y \text{ is } y \][/tex]
### Step 3: Combine all the LCM components
We now combine the LCM of the coefficients and the LCM of the variables:
[tex]\[ \text{LCM} = 24 \cdot x^7 \cdot w^6 \cdot y \][/tex]
So, the least common multiple of the monomials [tex]\( 24x^4w^6y \)[/tex] and [tex]\( 12x^7w^3 \)[/tex] is:
[tex]\[ 24x^7w^6y \][/tex]
Thus, the LCM of the expressions is:
[tex]\[ 24x^7w^6y \][/tex]
