Final answer:
The least common multiple (LCM) of 48x⁵y⁴ and 8x⁵y³ is 48x⁵y⁴, as it includes the highest powers of the variables and the largest coefficient that is a multiple of both original coefficients.
Step-by-step explanation:
The least common multiple (LCM) of two algebraic expressions is the smallest expression that is a multiple of both original expressions. Finding the LCM of 48x⁵y⁴ and 8x⁵y³ involves identifying the highest powers of the variable factors and the largest coefficient that is a multiple of both coefficients in the original expressions.
Steps to Find the LCM:
Identify the coefficients: 48 and 8. The LCM is 48 because 48 is a multiple of 8.
Identify the highest power of x: x⁵ is common in both, so x⁵ will be in the LCM.
Identify the highest power of y: y⁴ from the first expression is the highest, so y⁴ will be in the LCM.
Combining these, the LCM is 48x⁵y⁴.
No new multiplication or changes to the exponents are necessary because the terms in the second expression are already included in the LCM determined from the first expression.