Evaluate `root3(4096)`

Options

A. 19
B. 17
C. 18
D. 16

Answer

To evaluate the cube root of 4096, we can follow these steps:

### Step 1: Identify the prime factorization of 4096
First, we need to find the prime factorization 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

So, 4096 can be expressed as \( 2^{12} \) (since we divided by 2 a total of 12 times).

### Step 2: Use the property of cube roots
The cube root of a number can be expressed using exponents. Specifically, the cube root of \( a^b \) is \( a^{b/3} \).

Thus, we can write:
\[ \sqrt[3]{4096} = \sqrt[3]{2^{12}} = 2^{12/3} \]

### Step 3: Simplify the exponent
Now, simplify the exponent:
\[ 2^{12/3} = 2^4 \]

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

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

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