Find the value of `root(3)(-13824)`.

Options

A. 38
B. -38
C. 24
D. -24

Answer

To find the value of \(\sqrt[3]{-13824}\), we can follow these steps:

### Step 1: Factor the number
We start by factoring \(-13824\) into its prime factors. Since we are looking for the cube root, we can express \(-13824\) as a product of cubes.

\[ -13824 = -1 \times 13824 \]

### Step 2: Factor 13824
Next, we need to factor \(13824\). We can do this by dividing by small prime numbers:

\[ 13824 \div 2 = 6912 \\ 6912 \div 2 = 3456 \\ 3456 \div 2 = 1728 \\ 1728 \div 2 = 864 \\ 864 \div 2 = 432 \\ 432 \div 2 = 216 \\ 216 \div 2 = 108 \\ 108 \div 2 = 54 \\ 54 \div 2 = 27 \\ 27 \div 3 = 9 \\ 9 \div 3 = 3 \\ 3 \div 3 = 1 \]

Thus, the prime factorization of \(13824\) is:

\[ 13824 = 2^{11} \times 3^3 \]

### Step 3: Combine with the negative sign
Now, including the negative sign, we have:

\[ -13824 = -1 \times 2^{11} \times 3^3 \]

### Step 4: Express as a cube
We can express this in a form suitable for taking the cube root:

\[ -13824 = -1 \times (2^3)^3 \times 3^3 \times 2^2 \]

This simplifies to:

\[ -13824 = -1 \times (2^3 \times 3)^3 \times 2^2 \]

### Step 5: Calculate the cube root
Now, we can take the cube root:

\[ \sqrt[3]{-13824} = \sqrt[3]{-1} \times \sqrt[3]{(2^3 \times 3)^3} \times \sqrt[3]{2^2} \]

Calculating each part:

\[ \sqrt[3]{-1} = -1 \\ \sqrt[3]{(2^3 \times 3)^3} = 2 \times 3 = 6 \\ \sqrt[3]{2^2} = 2^{2/3} \]

However, since we are interested in the integer cube root, we can ignore the fractional part for now and focus on the main cube root:

\[ \sqrt[3]{-13824} = -6 \times 2^{2/3} \]

But we can also simplify this directly:

\[ \sqrt[3]{-13824} = -24 \]

### Final Answer
Thus, the value of \(\sqrt[3]{-13824}\) is:

\[ \boxed{-24} \]