Find the cube-roots of :
3375

Answer

To find the cube root of 3375, we will follow these steps:

### Step 1: Set the number
Let \( a = 3375 \).

### Step 2: Factor the number
We will factor 3375 into its prime factors.

1. Divide by 3:
- \( 3375 \div 3 = 1125 \)
2. Divide 1125 by 3:
- \( 1125 \div 3 = 375 \)
3. Divide 375 by 3:
- \( 375 \div 3 = 125 \)
4. Now, 125 is not divisible by 3, so we divide by 5:
- \( 125 \div 5 = 25 \)
5. Divide 25 by 5:
- \( 25 \div 5 = 5 \)
6. Finally, divide 5 by 5:
- \( 5 \div 5 = 1 \)

So, the prime factorization of 3375 is:
\[ 3375 = 3^3 \times 5^3 \]

### Step 3: Use the cube root property
We want to find \( a^{1/3} \):
\[ a^{1/3} = (3^3 \times 5^3)^{1/3} \]

### Step 4: Apply the property of exponents
Using the property \( (a^m \times b^m)^{n} = a^{m \cdot n} \times b^{m \cdot n} \):
\[ (3^3 \times 5^3)^{1/3} = 3^{3 \cdot \frac{1}{3}} \times 5^{3 \cdot \frac{1}{3}} \]
\[ = 3^1 \times 5^1 \]
\[ = 3 \times 5 \]

### Step 5: Calculate the final result
\[ 3 \times 5 = 15 \]

### Final Answer
The cube root of 3375 is \( 15 \).

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