Evaluate `root3(4096)`

Options

A. 18
B. 12
C. 14
D. 16

Answer

To evaluate the cube root of \( 4096 \), we will follow these steps:

### Step 1: Prime Factorization of 4096
We start by finding the prime factors of \( 4096 \). We can do this by dividing \( 4096 \) by \( 2 \) repeatedly until we reach \( 1 \).

- \( 4096 \div 2 = 2048 \)
- \( 2048 \div 2 = 1024 \)
- \( 1024 \div 2 = 512 \)
- \( 512 \div 2 = 256 \)
- \( 256 \div 2 = 128 \)
- \( 128 \div 2 = 64 \)
- \( 64 \div 2 = 32 \)
- \( 32 \div 2 = 16 \)
- \( 16 \div 2 = 8 \)
- \( 8 \div 2 = 4 \)
- \( 4 \div 2 = 2 \)
- \( 2 \div 2 = 1 \)

So, the prime factorization of \( 4096 \) is \( 2^{12} \).

### Step 2: Grouping the Factors
Next, we group the factors in sets of three because we are looking for the cube root.

Since \( 4096 = 2^{12} \), we can group the factors as follows:

\[ 2^{12} = (2^3)^4 = 8^4 \]

### Step 3: Finding the Cube Root
Now we can find the cube root of \( 4096 \):

\[ \sqrt[3]{4096} = \sqrt[3]{(2^3)^4} = 2^4 = 16 \]

### Final Answer
Thus, the cube root of \( 4096 \) is \( 16 \).

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