What is the LCM of p and q where `p =a^3b^2 and q=b^3 a^(2)?`

Answer

To find the LCM (Least Common Multiple) of \( p \) and \( q \) where \( p = a^3b^2 \) and \( q = b^3a^2 \), we can follow these steps:

### Step 1: Identify the prime factors
We start by identifying the prime factors in both expressions:
- For \( p = a^3b^2 \), the prime factors are \( a \) and \( b \).
- For \( q = b^3a^2 \), the prime factors are also \( a \) and \( b \).

### Step 2: Determine the highest powers of each prime factor
Next, we need to determine the highest power of each prime factor from both \( p \) and \( q \):
- For \( a \):
- In \( p \), the power of \( a \) is \( 3 \).
- In \( q \), the power of \( a \) is \( 2 \).
- The maximum power of \( a \) is \( \max(3, 2) = 3 \).

- For \( b \):
- In \( p \), the power of \( b \) is \( 2 \).
- In \( q \), the power of \( b \) is \( 3 \).
- The maximum power of \( b \) is \( \max(2, 3) = 3 \).

### Step 3: Write the LCM using the highest powers
Now we can write the LCM using the highest powers of each prime factor:
\[ \text{LCM}(p, q) = a^{\max(3, 2)} \cdot b^{\max(2, 3)} = a^3 \cdot b^3 \]

### Step 4: Final answer
Thus, the LCM of \( p \) and \( q \) is:
\[ \text{LCM}(p, q) = a^3b^3 \]