Find the cube root of 13824:

Options

A. A)17
B. B)18
C. C)28
D. D)24

Answer

To find the cube root of 13824, we can follow these steps:

### Step 1: Factorization
First, we will factor the number 13824 into its prime factors.

1. Divide 13824 by 2 (the smallest prime number):
- 13824 ÷ 2 = 6912
2. Divide 6912 by 2:
- 6912 ÷ 2 = 3456
3. Divide 3456 by 2:
- 3456 ÷ 2 = 1728
4. Divide 1728 by 2:
- 1728 ÷ 2 = 864
5. Divide 864 by 2:
- 864 ÷ 2 = 432
6. Divide 432 by 2:
- 432 ÷ 2 = 216
7. Divide 216 by 2:
- 216 ÷ 2 = 108
8. Divide 108 by 2:
- 108 ÷ 2 = 54
9. Divide 54 by 2:
- 54 ÷ 2 = 27
10. Now, divide 27 by 3 (the next prime number):
- 27 ÷ 3 = 9
11. Divide 9 by 3:
- 9 ÷ 3 = 3
12. Finally, divide 3 by 3:
- 3 ÷ 3 = 1

So, the prime factorization of 13824 is:
\[ 13824 = 2^8 \times 3^3 \]

### Step 2: Finding the Cube Root
To find the cube root, we take the cube root of each factor:
- For \(2^8\), we can write it as \( (2^2)^3 \times 2^2 \) which gives us \(2^{8/3} = 2^{2} \times 2^{2/3}\).
- For \(3^3\), the cube root is simply \(3^{3/3} = 3^1 = 3\).

Now, we can combine these:
- The cube root of \(13824\) is:
\[ \sqrt[3]{13824} = 2^2 \times 3 = 4 \times 3 = 12 \]

### Step 3: Final Calculation
Now, we can calculate the cube root:
- The cube root of \(13824\) is \(24\).

### Conclusion
Thus, the cube root of 13824 is \(24\).