Questions: Find the prime factorization of the number: 144 2^4 x 3^3 2^4 x 3^2 4^3 x 3^2 5^2 x 2^6 x 3^2 2^2 x 3^4

Transcript text: Find the prime factorization of the number: 144 $2^{4} \times 3^{3}$ $2^{4} \times 3^{2}$ $4^{3} \times 3^{2}$ $5^{2} \times 2^{6} \times 3^{2}$ $2^{2} \times 3^{4}$

Solution

Solution Steps

To find the prime factorization of the number 144, we need to repeatedly divide the number by its smallest prime factor until we are left with 1. The prime factors of 144 are 2 and 3. We will divide 144 by 2 until it is no longer divisible by 2, and then divide the result by 3 until it is no longer divisible by 3.

Step 1: Prime Factorization

To find the prime factorization of the number $ 144 $, we start by dividing it by the smallest prime number, which is $ 2 $. We continue dividing by $ 2 $ until it is no longer divisible.

\[ 144 \div 2 = 72 \\ 72 \div 2 = 36 \\ 36 \div 2 = 18 \\ 18 \div 2 = 9 \]

At this point, $ 9 $ is no longer divisible by $ 2 $.

Step 2: Continuing with the Next Prime Factor

Next, we divide $ 9 $ by the next smallest prime number, which is $ 3 $:

\[ 9 \div 3 = 3 \\ 3 \div 3 = 1 \]

Now we have reached $ 1 $, indicating that we have completed the factorization process.

Step 3: Compiling the Prime Factors

From the divisions, we can summarize the prime factors and their respective powers: