Let's solve each part of the question step-by-step:
### Part 1: Simplify [tex]\( 4x \cdot 12x^2 \)[/tex]
We need to multiply two algebraic expressions:
[tex]\[ 4x \cdot 12x^2 \][/tex]
First, separate the constants and the variables:
- Constants: [tex]\(4 \)[/tex] and [tex]\( 12\)[/tex]
- Variables: [tex]\( x \)[/tex] and [tex]\( x^2 \)[/tex]
Multiply the constants:
[tex]\[ 4 \times 12 = 48 \][/tex]
Next, multiply the variables using the laws of exponents ([tex]\( x \cdot x^2 = x^{1+2} = x^3 \)[/tex]):
[tex]\[ x \cdot x^2 = x^3 \][/tex]
Putting it all together:
[tex]\[ 4x \cdot 12x^2 = \boxed{48x^3} \][/tex]
### Part 2: Expression [tex]\( 6x + 3x^2 \)[/tex]
This expression is already in its simplest form as there are no common factors or like terms to combine:
[tex]\[ \boxed{6x + 3x^2} \][/tex]
### Part 3: Simplify [tex]\( 72 \cdot x^5 y^4 + 36 x^2 y^6 - 54 y^3 \)[/tex]
Let's look at the given expression:
[tex]\[ 72 \cdot x^5 y^4 + 36 x^2 y^6 - 54 y^3 \][/tex]
To factor out the greatest common factor (GCF) from each term, we need to find the GCF of the constants and the variables separately.
Constants:
- The constants are 72, 36, and 54. The GCF of 72, 36, and 54 is 18.
Variables:
- The variables in the terms are [tex]\( x^5 y^4\)[/tex], [tex]\( x^2 y^6\)[/tex] , and [tex]\( y^3\)[/tex]. The GCF for the variables would be the lowest power of each variable present in all terms:
- For [tex]\( x\)[/tex], the lowest power is 0 (since the [tex]\( y^3 \)[/tex] term does not include [tex]\( x \)[/tex]).
- For [tex]\( y \)[/tex], the lowest power is 3.
So, the GCF for the variables is [tex]\( y^3 \)[/tex].
Now, factor out [tex]\( 18y^3 \)[/tex] from the expression:
[tex]\[ 72x^5y^4 + 36x^2y^6 - 54y^3 = 18y^3 (4x^5y + 2x^2y^3 - 3) \][/tex]
Therefore, the simplified expression is:
[tex]\[ 18y^3 (4x^5y + 2x^2y^3 - 3) \][/tex]
So, the answer to Part 3 is:
[tex]\[ \boxed{18y^3 (4x^5y + 2x^2y^3 - 3)} \][/tex]
### Summary
1. [tex]\( 4x \cdot 12x^2 = 48x^3 \)[/tex]
2. [tex]\( 6x + 3x^2 \)[/tex]
3. [tex]\( 72 \cdot x^5 y^4 + 36x^2y^6 - 54y^3 = 18y^3 (4x^5y + 2x^2y^3 - 3) \)[/tex]
