LCM of Algebraic Expressions
Find the LCM of algebraic expressions in Grade 9 algebra. Factor each expression, take the highest power of each unique factor, and build the least common multiple for rational operations.
Key Concepts
Property To find the LCM of monomials, find the LCM of the coefficients. Then, for each variable, take the highest power that appears in any of the terms. The LCM is the product of the numerical LCM and the variables raised to their highest powers.
Explanation This is just like our playlist example, but now we're adding different genres, which are the variables! Treat each variable like a unique artist. For your final LCM, you need the LCM of the numbers, or coefficients, and the "greatest hits" version of each artist, which means taking the highest power of each variable you can find.
Examples Find the LCM of $10a^3c^5$ and $15a^2c^4$. The LCM of 10 and 15 is 30. The highest power of $a$ is $a^3$ and of $c$ is $c^5$. So, the LCM is $30a^3c^5$. Find the LCM of $6p^2s^3$ and $8m^3p$. The LCM of 6 and 8 is 24. The highest powers are $p^2$, $s^3$, and $m^3$. The LCM is $24m^3p^2s^3$.
Common Questions
How do you find the LCM of algebraic expressions?
Factor each expression completely. Take each unique factor at its highest power that appears across all expressions. Multiply those factors together to get the LCM.
Why do you need the LCM when adding algebraic fractions?
To add fractions with unlike denominators, you need a common denominator. The LCM is the LCD — the smallest common denominator, keeping expressions as simple as possible.
What is the difference between LCM of numbers and LCM of algebraic expressions?
For numbers use prime factorization; for algebraic expressions factor polynomials and include variable factors too. Take the highest power of each unique factor from all expressions.