Let's simplify each term separately, step-by-step, multiplying all the factors together.
### Term 1: [tex]\( 3x(2x^2) \)[/tex]
1. Identify the factors: [tex]\( 3 \)[/tex], [tex]\( x \)[/tex], [tex]\( 2 \)[/tex], and [tex]\( x^2 \)[/tex].
2. Multiply the constants: [tex]\( 3 \cdot 2 = 6 \)[/tex].
3. Simplify the variables:
- We have [tex]\( x \)[/tex] and [tex]\( x^2 \)[/tex].
- Recall the rule [tex]\( x^a \cdot x^b = x^{a+b} \)[/tex].
- Thus, [tex]\( x \cdot x^2 = x^{1+2} = x^3 \)[/tex].
4. Combine the simplified constants and variables: [tex]\( 6x^3 \)[/tex].
So, [tex]\( 3x(2x^2) \)[/tex] simplifies to [tex]\( 24x^3 \)[/tex].
### Term 2: [tex]\( 3x(4x) \)[/tex]
1. Identify the factors: [tex]\( 3 \)[/tex], [tex]\( x \)[/tex], [tex]\( 4 \)[/tex], and [tex]\( x \)[/tex].
2. Multiply the constants: [tex]\( 3 \cdot 4 = 12 \)[/tex].
3. Simplify the variables:
- We have [tex]\( x \)[/tex] and another [tex]\( x \)[/tex].
- Using the rule [tex]\( x^a \cdot x^b = x^{a+b} \)[/tex],
- [tex]\( x \cdot x = x^{1+1} = x^2 \)[/tex].
4. Combine the simplified constants and variables: [tex]\( 12x^2 \)[/tex].
So, [tex]\( 3x(4x) \)[/tex] simplifies to [tex]\( 12x^2 \)[/tex].
### Term 3: [tex]\( 3x(9) \)[/tex]
1. Identify the factors: [tex]\( 3 \)[/tex], [tex]\( x \)[/tex], and [tex]\( 9 \)[/tex].
2. Multiply the constants: [tex]\( 3 \cdot 9 = 27 \)[/tex].
3. The variable [tex]\( x \)[/tex] remains as [tex]\( x \)[/tex] since there are no other [tex]\( x \)[/tex] terms to combine with.
4. Combine the simplified constants and the variable: [tex]\( 27x \)[/tex].
So, [tex]\( 3x(9) \)[/tex] simplifies to [tex]\( 27x \)[/tex].
### Summary
After simplifying each term separately, we have:
[tex]\[
3 x\left(2 x^2\right) = 24x^3
\][/tex]
[tex]\[
3 x(4 x) = 12x^2
\][/tex]
[tex]\[
3 x(9) = 27x
\][/tex]
