`2root(3)(40)-4root(3)(320)+3root(3)(625)-3root(3)(5)` is equal to

Options

A. a) `-2root3(340)`
B. b) `0`
C. c) `root3(340)`
D. d) `root3(660)`

Answer

To simplify the expression \( 2\sqrt[3]{40} - 4\sqrt[3]{320} + 3\sqrt[3]{625} - 3\sqrt[3]{5} \), we will break down each term step by step.

### Step 1: Break down the cube roots
First, we will simplify each cube root term in the expression.

1. **For \( \sqrt[3]{40} \)**:
\[ 40 = 8 \times 5 \quad \text{(since \( 8 = 2^3 \))} \]
Therefore,
\[ \sqrt[3]{40} = \sqrt[3]{8 \times 5} = \sqrt[3]{8} \times \sqrt[3]{5} = 2\sqrt[3]{5} \]

2. **For \( \sqrt[3]{320} \)**:
\[ 320 = 64 \times 5 = 8 \times 40 = 8 \times 8 \times 5 \]
Therefore,
\[ \sqrt[3]{320} = \sqrt[3]{64 \times 5} = \sqrt[3]{64} \times \sqrt[3]{5} = 4\sqrt[3]{5} \]

3. **For \( \sqrt[3]{625} \)**:
\[ 625 = 125 \times 5 = 5^3 \times 5 = 5^4 \]
Therefore,
\[ \sqrt[3]{625} = \sqrt[3]{5^4} = 5^{4/3} = 5 \cdot \sqrt[3]{5} \]

4. **For \( \sqrt[3]{5} \)**:
This term remains as it is.

### Step 2: Substitute back into the expression
Now we substitute these simplified terms back into the original expression:
\[ 2\sqrt[3]{40} - 4\sqrt[3]{320} + 3\sqrt[3]{625} - 3\sqrt[3]{5} \]
becomes:
\[ 2(2\sqrt[3]{5}) - 4(4\sqrt[3]{5}) + 3(5\sqrt[3]{5}) - 3\sqrt[3]{5} \]
which simplifies to:
\[ 4\sqrt[3]{5} - 16\sqrt[3]{5} + 15\sqrt[3]{5} - 3\sqrt[3]{5} \]

### Step 3: Combine like terms
Now we combine the coefficients of \( \sqrt[3]{5} \):
\[ (4 - 16 + 15 - 3)\sqrt[3]{5} = (4 - 16 + 15 - 3)\sqrt[3]{5} = 0\sqrt[3]{5} \]

### Final Result
Thus, the entire expression simplifies to:
\[ 0 \]