`3root(3)(32)-5root(3)(4)+root(3)(500)` is equal to

Options

A. `4root(3)(6)`
B. `3sqrt(24)`
C. `6root(3)(4)`
D. 916

Answer

To solve the expression \( 3 \sqrt[3]{32} - 5 \sqrt[3]{4} + \sqrt[3]{500} \), we will simplify each term step by step.

### Step 1: Simplify \( \sqrt[3]{32} \)
We can express \( 32 \) as \( 8 \times 4 \):
\[ \sqrt[3]{32} = \sqrt[3]{8 \times 4} = \sqrt[3]{8} \times \sqrt[3]{4} \]
Since \( \sqrt[3]{8} = 2 \):
\[ \sqrt[3]{32} = 2 \sqrt[3]{4} \]

### Step 2: Substitute back into the expression
Now substitute \( \sqrt[3]{32} \) back into the original expression:
\[ 3 \sqrt[3]{32} = 3 \times 2 \sqrt[3]{4} = 6 \sqrt[3]{4} \]
So, we have:
\[ 6 \sqrt[3]{4} - 5 \sqrt[3]{4} + \sqrt[3]{500} \]

### Step 3: Combine like terms
Now combine the terms involving \( \sqrt[3]{4} \):
\[ (6 - 5) \sqrt[3]{4} + \sqrt[3]{500} = 1 \sqrt[3]{4} + \sqrt[3]{500} = \sqrt[3]{4} + \sqrt[3]{500} \]

### Step 4: Simplify \( \sqrt[3]{500} \)
We can express \( 500 \) as \( 125 \times 4 \):
\[ \sqrt[3]{500} = \sqrt[3]{125 \times 4} = \sqrt[3]{125} \times \sqrt[3]{4} \]
Since \( \sqrt[3]{125} = 5 \):
\[ \sqrt[3]{500} = 5 \sqrt[3]{4} \]

### Step 5: Substitute back into the expression
Now substitute \( \sqrt[3]{500} \) back into the expression:
\[ \sqrt[3]{4} + 5 \sqrt[3]{4} = (1 + 5) \sqrt[3]{4} = 6 \sqrt[3]{4} \]

### Final Answer
Thus, the expression \( 3 \sqrt[3]{32} - 5 \sqrt[3]{4} + \sqrt[3]{500} \) simplifies to:
\[ \boxed{6 \sqrt[3]{4}} \]