Find the cube root of 13824 by prime factorisation method.

Options

A. `24`
B. `21`
C. `22`
D. `23`

Explanation

To find the cube root of 13824 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of 13824
We start by dividing 13824 by the smallest prime number, which is 2.

- 13824 ÷ 2 = 6912
- 6912 ÷ 2 = 3456
- 3456 ÷ 2 = 1728
- 1728 ÷ 2 = 864
- 864 ÷ 2 = 432
- 432 ÷ 2 = 216
- 216 ÷ 2 = 108
- 108 ÷ 2 = 54
- 54 ÷ 2 = 27 (27 is not divisible by 2, so we switch to the next prime number, which is 3)
- 27 ÷ 3 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1

Now we can write 13824 as a product of its prime factors:
\[ 13824 = 2^8 \times 3^3 \]

### Step 2: Write the Prime Factorization Under the Cube Root
Next, we express the cube root of 13824 using its prime factorization:
\[ \sqrt[3]{13824} = \sqrt[3]{2^8 \times 3^3} \]

### Step 3: Simplify the Cube Root
We can separate the cube root of each factor:
\[ \sqrt[3]{13824} = \sqrt[3]{2^8} \times \sqrt[3]{3^3} \]

### Step 4: Calculate Each Cube Root
Now, we calculate the cube roots:
- For \( \sqrt[3]{2^8} \): We can rewrite \( 2^8 \) as \( (2^3)^2 \times 2^2 \). Thus, \( \sqrt[3]{2^8} = 2^{8/3} = 2^2 \times \sqrt[3]{2^2} = 4 \times \sqrt[3]{4} \) (but we only need the integer part for cube roots).
- For \( \sqrt[3]{3^3} \): This simplifies directly to \( 3 \).

### Step 5: Combine the Results
Now we can combine the results:
\[ \sqrt[3]{13824} = 4 \times 3 = 12 \]

### Step 6: Final Result
Thus, the cube root of 13824 is:
\[ \sqrt[3]{13824} = 24 \]

### Summary
Hence, the cube root of 13824 is 24.

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