Sure, let's work through these problems step by step!
### Problem 1: Simplifying [tex]\( 4x \cdot 12x^2 \)[/tex]
We need to multiply [tex]\( 4x \)[/tex] with [tex]\( 12x^2 \)[/tex].
1. First, multiply the coefficients: [tex]\( 4 \cdot 12 = 48 \)[/tex].
2. Then, multiply the variables: [tex]\( x \cdot x^2 = x^{1+2} = x^3 \)[/tex].
Thus, [tex]\( 4x \cdot 12x^2 = 48x^3 \)[/tex].
### Problem 2: Simplifying [tex]\( 6x + 3x^2 \)[/tex]
In this case, we need to factor the expression:
1. The common factor in both terms is [tex]\( 3x \)[/tex].
2. Factoring out [tex]\( 3x \)[/tex] from [tex]\( 6x + 3x^2 \)[/tex]:
- [tex]\( 6x = 3x \cdot 2 \)[/tex].
- [tex]\( 3x^2 = 3x \cdot x \)[/tex].
Hence, [tex]\( 6x + 3x^2 = 3x(2 + x) \)[/tex].
### Problem 3: Simplifying [tex]\( 27x^5y^4 + 36x^2y^6 - 54^3y^5 \)[/tex]
Let's simplify this expression step by step.
1. Observe [tex]\( 27x^5y^4 + 36x^2y^6 \)[/tex] first:
- Both terms have [tex]\( y^4 \)[/tex] as a common factor.
- Additionally, 27 and 36 share a factor of 9.
So, we can factor out [tex]\( 9y^4 \)[/tex]:
- [tex]\( 27x^5y^4 = 9y^4 \cdot 3x^5 \)[/tex]
- [tex]\( 36x^2y^6 = 9y^4 \cdot 4x^2 y^2 \)[/tex]
Now rewrite the expression: [tex]\( 9y^4(3x^5 + 4x^2y^2) \)[/tex].
2. Subtract the term involving [tex]\( 54^3y^5 \)[/tex]:
- Note that [tex]\( 54^3 = 157464 \)[/tex].
Thus, the term [tex]\( 54^3y^5 = 157464y^5 \)[/tex].
3. Since [tex]\( 157464y^5 \)[/tex] cannot be simplified with the previously grouped terms, we include it in the final expression:
So, combining all terms, we get:
[tex]\[ 27x^5y^4 + 36x^2y^6 - 157464y^5 \][/tex]
Factored form:
[tex]\[ y^4(27x^5 + 36x^2y^2 - 157464y) \][/tex]
### Final Results:
1. [tex]\( 4x \cdot 12x^2 = 48x^3 \)[/tex]
2. [tex]\( 6x + 3x^2 = 3x(x + 2) \)[/tex]
3. [tex]\( 27x^5y^4 + 36x^2y^6 - 54^3y^5 = y^4(27x^5 + 36x^2y^2 - 157464y) \)[/tex]
This concludes the detailed step-by-step solutions for the given problems.
