`root3(5832)=?`

Options

A. 22
B. 18
C. 16
D. 14

Answer

To find the cube root of 5832, we will follow these steps:

### Step 1: Prime Factorization
First, we need to break down the number 5832 into its prime factors.

- Divide 5832 by 2:
- 5832 ÷ 2 = 2916
- Divide 2916 by 2:
- 2916 ÷ 2 = 1458
- Divide 1458 by 2:
- 1458 ÷ 2 = 729
- Now, 729 is not divisible by 2, so we try the next prime number, which is 3:
- 729 ÷ 3 = 243
- 243 ÷ 3 = 81
- 81 ÷ 3 = 27
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

So, the prime factorization of 5832 is:
\[ 5832 = 2^3 \times 3^6 \]

### Step 2: Applying the Cube Root
Next, we apply the cube root to the prime factorization:
\[ \sqrt[3]{5832} = \sqrt[3]{2^3 \times 3^6} \]

Using the property of cube roots, we can separate the factors:
\[ \sqrt[3]{2^3} \times \sqrt[3]{3^6} \]

### Step 3: Simplifying the Cube Roots
Now, we simplify each cube root:
- \( \sqrt[3]{2^3} = 2 \)
- \( \sqrt[3]{3^6} = 3^{6/3} = 3^2 = 9 \)

### Step 4: Multiply the Results
Now, we multiply the results from the cube roots:
\[ 2 \times 9 = 18 \]

### Final Answer
Thus, the cube root of 5832 is:
\[ \sqrt[3]{5832} = 18 \]

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