When completed simplified, the expression 3𝑥(4𝑥−6)2(3𝑥)2(3x)3x(4x−6)​ is equivalent to

To simplify the expression $3x(4x - 6)^2(3x)^2(3x)^3$, we can follow these steps:

1. Break Down the Problem

Identify and organize the components of the expression:

• $3x$
• $(4x - 6)^2$
• $(3x)^2$
• $(3x)^3$

2. Relevant Concepts

We will use the rules of exponents and polynomial expansion for simplification.

3. Analysis and Detail

1. Expand Each Term

   • The term $(3x)^2$ can be simplified to $9x^2$.
   • The term $(3x)^3$ simplifies to $27x^3$.
   • The term $(4x - 6)^2$ can be expanded using the formula $(a - b)^2 = a^2 - 2ab + b^2$: $(4x - 6)^2 = (4x)^2 - 2 \cdot 4x \cdot 6 + 6^2 = 16x^2 - 48x + 36$

2. Combine All Terms Substitute the expansions back into the expression: $3x(16x^2 - 48x + 36)(9x^2)(27x^3)$

3. Combine the Coefficients

   • Multiply the coefficients: $3 \cdot 9 \cdot 27 = 729$

4. Combine the Powers of $x$ $x \cdot x^2 \cdot x^3 = x^{1 + 2 + 3} = x^6$

4. Verify and Summarize

After expanding and combining all terms, the form would be: $729x^6(16x^2 - 48x + 36)$

Final Answer

The simplified expression is: $729x^6(16x^2 - 48x + 36)$