Find the cube-roots of :
4096

Answer

To find the cube root of 4096, we will follow these steps:

### Step 1: Factor the number 4096
We start by finding the prime factors of 4096. We can divide 4096 by 2 repeatedly until we reach 1.

- 4096 ÷ 2 = 2048
- 2048 ÷ 2 = 1024
- 1024 ÷ 2 = 512
- 512 ÷ 2 = 256
- 256 ÷ 2 = 128
- 128 ÷ 2 = 64
- 64 ÷ 2 = 32
- 32 ÷ 2 = 16
- 16 ÷ 2 = 8
- 8 ÷ 2 = 4
- 4 ÷ 2 = 2
- 2 ÷ 2 = 1

Thus, the prime factorization of 4096 is:
\[ 4096 = 2^{12} \]

### Step 2: Apply the cube root formula
Now that we have the prime factorization, we can express the cube root of 4096 using the formula:
\[ \sqrt[3]{a} = a^{1/3} \]

So, we have:
\[ \sqrt[3]{4096} = \sqrt[3]{2^{12}} \]

### Step 3: Simplify using the power of a power rule
Using the property of exponents, \( (a^m)^n = a^{m \cdot n} \), we can simplify:
\[ \sqrt[3]{2^{12}} = 2^{12 \cdot \frac{1}{3}} = 2^{\frac{12}{3}} = 2^4 \]

### Step 4: Calculate \( 2^4 \)
Now we calculate \( 2^4 \):
\[ 2^4 = 16 \]

### Final Answer
Thus, the cube root of 4096 is:
\[ \sqrt[3]{4096} = 16 \]

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