Answered

Which of the following shows the prime factorization of 200 using exponential notation?


A$2^3 \cdot 5^2$
B$2 \cdot 10^2$
C$2^2 \cdot 5^2$
D$8 \cdot 25$

Verified Answer :

  • Divide 200 repeatedly by its smallest prime factors: \(200 \div 2 = 100\), \(100 \div 2 = 50\), \(50 \div 2 = 25\), and \(25 \div 5 = 5\), \(5 \div 5 = 1\).
  • Group the identical factors: \(2 \times 2 \times 2 = 2^3\) and \(5 \times 5 = 5^2\).
  • Combine the powers to represent the number as the product of its prime factors: \(2^3 \cdot 5^2\).
  • Identify the correct option from the given list: \(\boxed{2^3 \cdot 5^2}\).

Explanation

  1. Understanding Prime Factorization

The task is to find the prime factorization of the integer 200 expressed in exponential notation. A prime factorization represents an integer as the product of its prime factors raised to their respective powers. The number 200 can be broken down into prime factors by repeatedly dividing by the smallest possible prime numbers.

  1. Performing Prime Factorization

To factor 200, we start by dividing by the smallest prime factor, which is 2:
\[200 \div 2 = 100\]
Next, divide the result, 100, by 2 again:
\[100 \div 2 = 50\]
Then, divide 50 by 2 again:
\[50 \div 2 = 25\]
The number 25 is no longer divisible by 2. We move to the next prime number, 5:
\[25 \div 5 = 5\]
Finally, divide 5 by 5:
\[5 \div 5 = 1\]

  1. Writing in Exponential Notation

From the divisions above, we have identified the factors: \(200 = 2 \times 2 \times 2 \times 5 \times 5\).
Grouping these factors, we can write them in exponential notation:

  • There are three 2s, so we write this as \(2^3\).
  • There are two 5s, so we write this as \(5^2\).

Therefore, the prime factorization of 200 is \(2^3 \cdot 5^2\).

  1. Evaluating the Options and Final Result

We now compare our result, \(2^3 \cdot 5^2\), with the given options:

  • \(2^3 \cdot 5^2 = 8 \cdot 25 = 200\) (Correct and uses only prime factors)
  • \(2 \cdot 10^2 = 2 \cdot 100 = 200\) (Not a prime factorization as 10 is composite)
  • \(2^2 \cdot 5^2 = 4 \cdot 25 = 100 eq 200\) (Incorrect value)
  • \(8 \cdot 25 = 200\) (Not a prime factorization as 8 and 25 are composite)

The only correct prime factorization among the options is \(2^3 \cdot 5^2\), which is \(\boxed{2^3 \cdot 5^2}\).

Examples

Prime factorization is used in cryptography and computer science to ensure secure communications. For instance, creating a digital signature or encrypting a message often depends on the difficulty of finding the prime factors of very large composite numbers. By understanding how to break down smaller numbers like 200 into their fundamental building blocks (\(2^3 \cdot 5^2\)), you build a strong foundation for exploring these complex algorithms that protect your data online.

Related Questions