- The expression \(2^5 \cdot 11^4 \cdot 9^4\) is not a prime factorization because 9 is not a prime number; therefore, option B is correct.
- To find the true prime factorization, replace 9 with \(3^2\): \(2^5 \cdot 11^4 \cdot (3^2)^4\).
- Apply the power rule \((a^m)^n = a^{m \cdot n}\) to simplify: \(2^5 \cdot 11^4 \cdot 3^8\).
- The final prime factorization is \(\boxed{2^5 \cdot 3^8 \cdot 11^4}\).
Explanation
- Problem Analysis and Identifying the Issue
The expression \(2^5 \cdot 11^4 \cdot 9^4\) is a product of three terms: \(2^5\), \(11^4\), and \(9^4\). For an expression to be a prime factorization, every base of the powers must be a prime number. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. In our expression, the bases are 2, 11, and 9. We observe that 2 is prime and 11 is prime, but 9 is not prime because it is a composite number: \(9 = 3 \times 3 = 3^2\). Therefore, the expression is not a prime factorization because not all of its factors are prime numbers. This corresponds to option B.
- Finding the Prime Factorization
To find the prime factorization of the expression, we must express all bases as prime numbers. We replace the base 9 with \(3^2\). Substituting this into the original expression, we get:
\[2^5 \cdot 11^4 \cdot (3^2)^4\]
Using the power of a power rule for exponents, \((a^m)^n = a^{m \cdot n}\), we simplify \((3^2)^4\) to \(3^{2 \times 4} = 3^8\).
The expression now becomes:
\[2^5 \cdot 11^4 \cdot 3^8\]
Reordering the factors, we obtain the prime factorization:
\[2^5 \cdot 3^8 \cdot 11^4\]
Since 2, 3, and 11 are all prime numbers, this is the prime factorization of the given number.
- Final Answer and Summary
The reason the original expression was not a prime factorization is that the factor 9 is composite. By decomposing 9 into \(3^2\), we reached the final prime factorization. The final answer for the prime factorization is \(\boxed{2^5 \cdot 3^8 \cdot 11^4}\) and the correct choice is B.
Examples
Prime factorization is a fundamental tool in mathematics used to break down large numbers into their "building blocks." In real life, it is essential for modern cybersecurity, specifically in RSA encryption, where the security of digital data relies on the difficulty of factoring very large composite numbers into their two original large prime factors. By understanding prime factorization, you are learning the foundation upon which secure online communication is built! For example, if you wanted to find the common factors of two large numbers to simplify a fraction, you would first find their prime factorizations, such as \(12 = 2^2 \cdot 3^1\) and \(18 = 2^1 \cdot 3^2\), and then determine their greatest common divisor easily.