`root3(512)=?`

Options

A. 6
B. 7
C. 8
D. 9

Answer

To find the cube root of 512, we can follow these steps:

### Step-by-Step Solution:

1. **Factor 512 into Prime Factors:**
- Start dividing 512 by 2 (the smallest prime number) until you cannot divide anymore.
- 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, the prime factorization of 512 is:
\[ 512 = 2^9 \]

2. **Group the Factors:**
- Since we are looking for the cube root, we group the factors in sets of three (because we are taking the cube root).
- From \(2^9\), we can group it as:
\[ (2^3) \times (2^3) \times (2^3) = 2^9 \]

3. **Rewrite in Terms of Cube:**
- Each group of \(2^3\) is equal to 8, so we can rewrite \(512\) as:
\[ 512 = 8^3 \]

4. **Take the Cube Root:**
- Now, we can find the cube root of \(512\):
\[ \sqrt[3]{512} = \sqrt[3]{8^3} = 8 \]

### Final Answer:
\[ \sqrt[3]{512} = 8 \]

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