Finding the LCM of Algebraic Expressions: ab²c², a²bc, and a³b³c²

To find the Least Common Multiple (LCM) of algebraic expressions, we need to identify all the unique prime factors (variables in this case) present in the expressions and take the highest power of each factor.

Let's break down the given expressions:

• The first expression is $ab^2c^2$. This can be written as $a^1 \cdot b^2 \cdot c^2$.
• The second expression is $a^2bc$. This can be written as $a^2 \cdot b^1 \cdot c^1$.
• The third expression is $a^3b^3c^2$. This can be written as $a^3 \cdot b^3 \cdot c^2$.

Now, let's identify the unique variables and their highest powers across all the expressions:

• Variable 'a': The powers are $a^1$ (from $ab^2c^2$), $a^2$ (from $a^2bc$), and $a^3$ (from $a^3b^3c^2$). The highest power of 'a' is $a^3$.
• Variable 'b': The powers are $b^2$ (from $ab^2c^2$), $b^1$ (from $a^2bc$), and $b^3$ (from $a^3b^3c^2$). The highest power of 'b' is $b^3$.
• Variable 'c': The powers are $c^2$ (from $ab^2c^2$), $c^1$ (from $a^2bc$), and $c^2$ (from $a^3b^3c^2$). The highest power of 'c' is $c^2$.

The LCM is found by multiplying the highest powers of all the unique variables together.

LCM $= (\text{Highest power of a}) \cdot (\text{Highest power of b}) \cdot (\text{Highest power of c})$

LCM $= a^3 \cdot b^3 \cdot c^2$

So, the LCM of $ab^2c^2$, $a^2bc$, and $a^3b^3c^2$ is $a^3b^3c^2$.

Revision Table: Understanding LCM of Algebraic Terms

Additional Information: LCM vs HCF for Algebraic Expressions

It is useful to compare the process of finding LCM with finding the Highest Common Factor (HCF) of algebraic expressions.

• LCM: Take the highest power of all unique factors present in any of the terms.
• HCF: Take the lowest power of only the common factors present in all the terms.

For the given expressions $ab^2c^2$, $a^2bc$, and $a^3b^3c^2$:

• Common factors are a, b, and c.
• Lowest power of 'a' is $a^1$.
• Lowest power of 'b' is $b^1$.
• Lowest power of 'c' is $c^1$.

So, the HCF of $ab^2c^2$, $a^2bc$, and $a^3b^3c^2$ would be $a^1b^1c^1 = abc$. This is different from the LCM.

Understanding the distinction between using the highest power for LCM and the lowest power for HCF is key when working with algebraic expressions.