Find the cube root of each of the following numbers by prime factorisation method. 91125

Answer

To find the cube root of 91125 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization
First, we need to perform the prime factorization of the number 91125. We will divide the number by the smallest prime numbers until we reach 1.

1. **Divide by 5**:
- \( 91125 \div 5 = 18225 \)
2. **Divide by 5 again**:
- \( 18225 \div 5 = 3645 \)
3. **Divide by 5 again**:
- \( 3645 \div 5 = 729 \)
4. **Divide by 3**:
- \( 729 \div 3 = 243 \)
5. **Divide by 3 again**:
- \( 243 \div 3 = 81 \)
6. **Divide by 3 again**:
- \( 81 \div 3 = 27 \)
7. **Divide by 3 again**:
- \( 27 \div 3 = 9 \)
8. **Divide by 3 again**:
- \( 9 \div 3 = 3 \)
9. **Divide by 3 again**:
- \( 3 \div 3 = 1 \)

So, the prime factorization of 91125 is:
\[ 91125 = 5^3 \times 3^6 \]

### Step 2: Grouping the Prime Factors
Next, we group the prime factors in sets of three since we are finding the cube root.

From the factorization:
- \( 5^3 \) contributes one group of 5.
- \( 3^6 \) can be grouped as \( (3^3) \times (3^3) \), contributing two groups of 3.

### Step 3: Taking the Cube Root
Now we can take the cube root of the grouped factors:
\[ \sqrt[3]{91125} = \sqrt[3]{5^3 \times (3^3) \times (3^3)} = 5 \times 3 \times 3 \]

### Step 4: Calculate the Result
Now we multiply the factors:
\[ 5 \times 3 \times 3 = 5 \times 9 = 45 \]

### Final Answer
Thus, the cube root of 91125 is:
\[ \sqrt[3]{91125} = 45 \]

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