Questions: Use the prime factorization to group the factors and write as a product of two factors. -6 k^10=(-1 cdot 3 cdot k cdot k cdot k cdot k cdot k cdot k cdot k) cdot(2 cdot k cdot k cdot k) -6 k^10=-3 k^7(square) (Type your answer using exponential notation.)

Use the prime factorization to group the factors and write as a product of two factors.

-6 k^10=(-1 cdot 3 cdot k cdot k cdot k cdot k cdot k cdot k cdot k) cdot(2 cdot k cdot k cdot k)

-6 k^10=-3 k^7(square)

(Type your answer using exponential notation.)

Transcript text: Use the prime factorization to group the factors and write as a product of two factors. \[ \begin{array}{l} -6 k^{10}=(-1 \cdot 3 \cdot k \cdot k \cdot k \cdot k \cdot k \cdot k \cdot k) \cdot(2 \cdot k \cdot k \cdot k) \\ -6 k^{10}=-3 k^{7}(\square) \end{array} \] (Type your answer using exponential notation.)

Solution

Solution Steps

To solve this problem, we need to express \(-6k^{10}\) as a product of two factors using its prime factorization. We start by breaking down \(-6k^{10}\) into its prime factors and then group them into two separate products. The first factor is given as \(-3k^7\), and we need to determine the second factor by dividing the original expression by the first factor.

Step 1: Prime Factorization of \(-6k^{10}\)

We start with the expression: \[ -6k^{10} = -1 \cdot 2 \cdot 3 \cdot k^{10} \] This can be rewritten as: \[ -6k^{10} = (-3k^7) \cdot (2k^3) \]

Step 2: Identifying the Factors

From the prime factorization, we can identify the two factors:

The first factor is given as: \[ -3k^7 \]
The second factor, calculated from the division of the original expression by the first factor, is: \[ 2k^3 \]

Step 3: Writing the Final Expression

Thus, we can express \(-6k^{10}\) as a product of the two factors: \[ -6k^{10} = (-3k^7) \cdot (2k^3) \]