What is the simplified value of ` 2 root(3)(243) + 3 root(3)(9) - root(3)(1125)?`

Options

A. `5sqrt(9)`
B. `4 sqrt(9)`
C. `7 sqrt(9)`
D. `11 root(3)(9)`

Answer

To simplify the expression \( 2 \sqrt[3]{243} + 3 \sqrt[3]{9} - \sqrt[3]{1125} \), we will follow these steps:

### Step 1: Simplify each cube root
1. **Calculate \( \sqrt[3]{243} \)**:
- \( 243 = 3^5 \)
- Therefore, \( \sqrt[3]{243} = \sqrt[3]{3^5} = 3^{5/3} = 3^{1 + 2/3} = 3 \cdot \sqrt[3]{9} \).

2. **Calculate \( \sqrt[3]{9} \)**:
- \( 9 = 3^2 \)
- Therefore, \( \sqrt[3]{9} = 3^{2/3} \).

3. **Calculate \( \sqrt[3]{1125} \)**:
- \( 1125 = 3^2 \times 5^3 \)
- Therefore, \( \sqrt[3]{1125} = \sqrt[3]{3^2 \cdot 5^3} = \sqrt[3]{3^2} \cdot \sqrt[3]{5^3} = 3^{2/3} \cdot 5 \).

### Step 2: Substitute back into the expression
Now substituting these values back into the original expression:

\[ 2 \sqrt[3]{243} = 2 \cdot (3 \cdot \sqrt[3]{9}) = 6 \sqrt[3]{9} \]

\[ 3 \sqrt[3]{9} = 3 \sqrt[3]{9} \]

\[ -\sqrt[3]{1125} = - (3^{2/3} \cdot 5) = -5 \cdot \sqrt[3]{9} \]

### Step 3: Combine like terms
Now we can combine the terms:

\[ 6 \sqrt[3]{9} + 3 \sqrt[3]{9} - 5 \sqrt[3]{9} = (6 + 3 - 5) \sqrt[3]{9} = 4 \sqrt[3]{9} \]

### Final Answer
Thus, the simplified value of the expression \( 2 \sqrt[3]{243} + 3 \sqrt[3]{9} - \sqrt[3]{1125} \) is:

\[ \boxed{4 \sqrt[3]{9}} \]