express 144 as a product of its prime numbers factors in index form

express 144 as a product of its prime numbers factors in index form

The correct answer and explanation is:

To express 144 as a product of its prime factors in index form, we first need to factorize it into prime numbers.

Step-by-Step Prime Factorization of 144

1. Start with 144.
   Since 144 is an even number, we can divide it by 2 (the smallest prime number): 144÷2=72144 \div 2 = 72
2. Factor 72.
   72 is also even, so we divide it by 2 again: 72÷2=3672 \div 2 = 36
3. Factor 36.
   36 is still even, so divide it by 2 once more: 36÷2=1836 \div 2 = 18
4. Factor 18.
   18 is even, so divide by 2: 18÷2=918 \div 2 = 9
5. Factor 9.
   9 is not even, but it can be divided by 3 (since 9 is divisible by 3): 9÷3=39 \div 3 = 3
6. Finally, factor 3.
   Since 3 is already a prime number, we stop here.

Prime Factorization in Index Form

Now, we can express the prime factorization of 144 by multiplying all the prime factors: 144=2×2×2×2×3×3144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3

In index form, this becomes: 144=24×32144 = 2^4 \times 3^2

Explanation

The process of factorizing involves breaking down a number into its smallest divisible prime factors. For 144, the prime factorization shows that it is made up of four 2’s and two 3’s. The exponents in the index form 24×322^4 \times 3^2 represent how many times each prime factor appears in the factorization. This method ensures that the number is expressed in terms of its prime building blocks, making it easier to work with, especially when simplifying expressions or solving problems involving powers or factors.