Let’s focus on simplifying the task of finding the prime factorizations step-by-step.
### 1) Prime Factorization of [tex]\( 16y \)[/tex]:
First, let's separate 16 and [tex]\( y \)[/tex]:
- The number 16 can be broken down into its prime factors.
- [tex]\( y \)[/tex] is already a prime number if it’s considered a variable.
Step-by-Step Factorization:
- We start with 16: [tex]\( 16 = 2 \times 8 \)[/tex]
- Factor 8 further: [tex]\( 8 = 2 \times 4 \)[/tex]
- Factor 4 further: [tex]\( 4 = 2 \times 2 \)[/tex]
So, combining all the factors, we get:
[tex]\[ 16 = 2 \times 2 \times 2 \times 2 = 2^4 \][/tex]
Thus, the prime factorization of [tex]\( 16y \)[/tex]:
[tex]\[ 16y = 2^4 \cdot y \][/tex]
### 2) Prime Factorization of [tex]\( 4v \)[/tex]:
First, let's separate 4 and [tex]\( v \)[/tex]:
- The number 4 can be broken down into its prime factors.
- [tex]\( v \)[/tex] is a prime number if it's considered a variable.
Step-by-Step Factorization:
- We start with 4: [tex]\( 4 = 2 \times 2 \)[/tex]
Thus, the prime factorization of [tex]\( 4v \)[/tex]:
[tex]\[ 4v = 2^2 \cdot v \][/tex]
### 3) Interpret and Correct Provided Expressions:
The given expressions seem disjointed and possibly incorrect, making it hard to interpret directly. However, let's confirm known prime factorizations before moving on to specific needs.
### 4) Prime Factorization of [tex]\( 21x^2 \)[/tex]:
First, let's separate 21 and [tex]\( x^2 \)[/tex]:
- The number 21 can be broken down into its prime factors.
- [tex]\( x^2 \)[/tex] is considered as [tex]\( x \cdot x \)[/tex].
Step-by-Step Factorization:
- We start with 21: [tex]\( 21 = 3 \times 7 \)[/tex]
Thus, the prime factorization of [tex]\( 21x^2 \)[/tex]:
[tex]\[ 21x^2 = 3 \times 7 \times x \times x \][/tex]
Recap of all the prime factorizations:
1. [tex]\( 16y = 2^4 \cdot y \)[/tex]
2. [tex]\( 4v = 2^2 \cdot v \)[/tex]
3. [tex]\( 21x^2 = 3 \times 7 \times x^2 \)[/tex]
Now, the problem seems to involve manipulation beyond basic prime factorization, potentially leading us to algebraic simplifications or fraction forms. But with the given terms taken into account, we reiterate:
- Factor each numeric element distinctly.
- Maintain each distinct algebraic element alongside to convey full intentions.
Let me know if you need specific breakdowns or assistance across further equations from initial expressions!
