Finding the cube root of 91125 through prime factorization involves breaking down the number into its prime factors. Here’s a detailed step-by-step guide:

Step 1: Prime Factorization
- Start dividing 91125 by the smallest prime number, which is 3.
- **91125 ÷ 3 = 30375**
- Continue dividing:
- **30375 ÷ 3 = 10125**
- **10125 ÷ 3 = 3375**
- **3375 ÷ 3 = 1125**
- **1125 ÷ 3 = 375**
- **375 ÷ 3 = 125**
- Now, 125 is not divisible by 3. Move to the next prime number, which is 5.
- **125 ÷ 5 = 25**
- **25 ÷ 5 = 5**
- **5 ÷ 5 = 1**

Step 2: Collecting Prime Factors
- The complete prime factorization of 91125 is:
- **91125 = 3^6 × 5^3**

Step 3: Finding the Cube Root
- To find the cube root, we take one-third of the powers of each prime factor:
- For \(3^6\): \( \frac{6}{3} = 2 \) → gives \( 3^2 = 9 \)
- For \(5^3\): \( \frac{3}{3} = 1 \) → gives \( 5^1 = 5 \)

Step 4: Multiplying the Results
- Now, multiply these results to find the cube root:
- **Cube root of 91125 = 3^2 × 5^1 = 9 × 5 = 45**

Conclusion
- Therefore, the cube root of 91125 is **45**.