Let's simplify the given expression step by step:

$\frac{12\left(a^2b\right)^2 \cdot b^3}{\left(2ab\right)^2}$

Step 1: Expand the squares in the numerator and denominator

Start by expanding both the numerator and the denominator:

• $\left(a^2b\right)^2 = a^{2 \times 2}b^2 = a^4b^2$
• $\left(2ab\right)^2 = 2^2 \cdot a^2 \cdot b^2 = 4a^2b^2$

Now, the expression becomes:

$\frac{12a^4b^2 \cdot b^3}{4a^2b^2}$

Step 2: Simplify powers of $b$ in the numerator

Since $b^2 \cdot b^3 = b^{2+3} = b^5$, we now have:

$\frac{12a^4b^5}{4a^2b^2}$

Step 3: Simplify the expression by canceling common factors

Now, let's cancel the common factors between the numerator and denominator:

• $a^4$ and $a^2$ can be simplified: $a^{4-2} = a^2$
• $b^5$ and $b^2$ can be simplified: $b^{5-2} = b^3$

This results in:

$\frac{12a^2b^3}{4}$

Step 4: Simplify the constants

Finally, divide $12$ by $4$:

$\frac{12}{4} = 3$

Thus, the simplified expression is:

$3a^2b^3$

Final Answer:

$3a^2b^3$

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Would you like more details or have any questions about the process?

Here are some related questions:

1. How do you handle powers of variables when simplifying algebraic expressions?
2. Can this method be applied to more complex expressions involving multiple variables and exponents?
3. How do you factorize algebraic expressions before simplifying them?
4. What happens when both the numerator and denominator have the same base and exponent?
5. How do you simplify fractions with both variables and constants in the numerator and denominator?

Tip: Always look for common factors in both the numerator and denominator before simplifying to make the process easier.