To find the least common multiple (LCM) of the two expressions [tex]\(12u^8x^7\)[/tex] and [tex]\(16u^4x^6y^3\)[/tex], we will follow these steps:
### Step 1: Identify the coefficients
The coefficients of the two expressions are:
- For [tex]\(12u^8x^7\)[/tex], the coefficient is 12.
- For [tex]\(16u^4x^6y^3\)[/tex], the coefficient is 16.
### Step 2: Find the LCM of the coefficients
The least common multiple (LCM) of 12 and 16 can be determined by finding the smallest number that both coefficients can divide without a remainder. In this case, the LCM of 12 and 16 is 48.
### Step 3: Determine the highest powers of each variable
Next, we consider the variables [tex]\(u\)[/tex], [tex]\(x\)[/tex], and [tex]\(y\)[/tex] in both expressions. The LCM expression will have the highest power of each variable present in the given expressions.
- For the variable [tex]\(u\)[/tex]:
- In [tex]\(12u^8x^7\)[/tex], [tex]\(u\)[/tex] is raised to the power of 8.
- In [tex]\(16u^4x^6y^3\)[/tex], [tex]\(u\)[/tex] is raised to the power of 4.
- The highest power of [tex]\(u\)[/tex] is 8.
- For the variable [tex]\(x\)[/tex]:
- In [tex]\(12u^8x^7\)[/tex], [tex]\(x\)[/tex] is raised to the power of 7.
- In [tex]\(16u^4x^6y^3\)[/tex], [tex]\(x\)[/tex] is raised to the power of 6.
- The highest power of [tex]\(x\)[/tex] is 7.
- For the variable [tex]\(y\)[/tex]:
- In [tex]\(12u^8x^7\)[/tex], [tex]\(y\)[/tex] is absent, so we treat it as [tex]\(y^0\)[/tex].
- In [tex]\(16u^4x^6y^3\)[/tex], [tex]\(y\)[/tex] is raised to the power of 3.
- The highest power of [tex]\(y\)[/tex] is 3.
### Step 4: Construct the LCM expression
Combining the LCM of the coefficients and the highest powers of each variable, we get the final LCM expression:
[tex]\[
48u^8x^7y^3
\][/tex]
### Summary
The least common multiple (LCM) of the expressions [tex]\(12u^8x^7\)[/tex] and [tex]\(16u^4x^6y^3\)[/tex] is:
[tex]\[
48u^8x^7y^3
\][/tex]
