Question

which of the following shows the prime factorization of 200 using exponential notation?
8·25
2²·5²
2³·5²
2·10²

Explanation:

Step1: Recall prime factorization

Prime factorization means expressing a number as a product of prime numbers. First, factorize 200.
200 divided by 2 is 100, 100 divided by 2 is 50, 50 divided by 2 is 25, 25 divided by 5 is 5, 5 divided by 5 is 1. So we have \(200 = 2\times2\times2\times5\times5\).

Step2: Use exponential notation

Exponential notation is used to represent repeated multiplication. For the prime factor 2, it appears 3 times, so \(2^3\). For the prime factor 5, it appears 2 times, so \(5^2\). Thus, the prime factorization of 200 in exponential notation is \(2^3\cdot5^2\).
Now check the options:

• \(8\cdot25\): 8 and 25 are not prime numbers, so this is not prime factorization.
• \(2^2\cdot5^2\): \(2^2 = 4\), \(4\times25 = 100

eq200\), incorrect.

• \(2^3\cdot5^2\): \(2^3 = 8\), \(5^2 = 25\), \(8\times25 = 200\), and 2 and 5 are primes, correct.
• \(2\cdot10^2\): 10 is not a prime number, so this is not prime factorization.

Answer:

\(2^3 \cdot 5^2\) (the option with \(2^3 \cdot 5^2\))