How do you express the prime factorization of the numbers 100 and 540 in exponential form?

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The prime factorization of a number involves breaking it down into its prime factors, which are the building blocks of all integers. To express these factors in exponential form, we count how many times each prime number appears in the factorization.

Prime Factorization of 100

Let’s start with the number 100:

  1. First, divide 100 by 2 (the smallest prime number).
    100 ÷ 2 = 50
  2. Next, divide 50 by 2 again.
    50 ÷ 2 = 25
  3. 25 is not divisible by 2, move to the next prime number, which is 3 (not applicable here). The next prime is 5.
    25 ÷ 5 = 5
  4. Lastly, divide 5 by 5.
    5 ÷ 5 = 1

So, the prime factors are:

  • 2 (appears 2 times)
  • 5 (appears 2 times)

Thus, the prime factorization of 100 in exponential form is:

100 = 22 × 52

Prime Factorization of 540

Now, let’s factor 540:

  1. Divide 540 by 2.
    540 ÷ 2 = 270
  2. Divide 270 by 2 again.
    270 ÷ 2 = 135
  3. 135 is not divisible by 2, so we check the next prime number, which is 3.
    135 ÷ 3 = 45
  4. Divide 45 by 3 again.
    45 ÷ 3 = 15
  5. Lastly, divide 15 by 3 one more time.
    15 ÷ 3 = 5
  6. And finally, divide 5 by 5.
    5 ÷ 5 = 1

So, the prime factors are:

  • 2 (appears 2 times)
  • 3 (appears 3 times)
  • 5 (appears 1 time)

Thus, the prime factorization of 540 in exponential form is:

540 = 22 × 33 × 51

In conclusion, the prime factorization of 100 is 22 × 52, and for 540, it is 22 × 33 × 51. This method helps simplify the numbers into their essential prime components, making them easier to work with in various mathematical contexts.

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