Question

below is one way to express the number 225 as product of prime factors. how is this prime factorization written in exponential notation? 225 is factored as 5, 5, 5 (wait, no, the factor tree: 225 splits into 5 and 45; 45 splits into 5 and 9; 9 splits into 3 and 3. so the prime factors are 5, 5, 5? wait no, looking at the bottom: 5 × 5 × 3 × 3? wait the image shows the bottom as 5 × 5 × 3 × 3? wait the users image: the factor tree has 225, then 5 and 45; 45 splits into 5 and 9; 9 splits into 3 and 3. then the bottom is 5 (from first branch) × 5 (from 45s first) × 3 × 3? wait the text below the tree is 5 × 5 × 3 × 3? then the options: a: 2×5×2×3, b: 45×9×5⁵×3², c: 5²×3², d: 5×5×3×3. select an answer: a, b, c, d options as given. so ocr text: below is one way to express the number 225 as product of prime factors. how is this prime factorization written in exponential notation? factor tree image select an answer a 225 = 2 × 5 × 2 × 3 b 225 = 45 × 9 × 5⁵ × 3² c 225 = 5² × 3² d 225 = 5 × 5 × 3 × 3

Explanation:

Step1: Identify prime factors

From the factor tree, the prime factors of 225 are 5, 5, 3, 3 (since \(225 = 5\times5\times3\times3\)).

Step2: Count exponents for each prime

For prime 5: it appears 2 times, so we write \(5^2\). For prime 3: it appears 2 times, so we write \(3^2\).

Step3: Write in exponential notation

Combining these, the prime factorization in exponential notation is \(225 = 5^2\times3^2\). Now we check the options:

• Option A: Incorrect, as 2 is not a factor of 225.
• Option B: Incorrect, as 45 and 9 are not prime factors, and the exponents for 5 is wrong.
• Option C: Correct, matches our calculation.
• Option D: Correct in prime factors but not in exponential notation (it's the expanded form, not exponential).

Answer:

C. \(225 = 5^2 \times 3^2\)