Simplify: ` 5root3(250)+7root3(16)-14root3(54)`

Options

A. `-2root3(2)`
B. `-3root3(2)`
C. `3root3(2)`
D. `2root3(2)`

Answer

To simplify the expression \( 5\sqrt[3]{250} + 7\sqrt[3]{16} - 14\sqrt[3]{54} \), we will follow these steps:

### Step 1: Simplify each cube root term

1. **Simplifying \( \sqrt[3]{250} \)**:
\[ 250 = 2 \times 5^3 \]
Thus,
\[ \sqrt[3]{250} = \sqrt[3]{2 \times 5^3} = \sqrt[3]{2} \times 5 \]

2. **Simplifying \( \sqrt[3]{16} \)**:
\[ 16 = 2^4 = 2^3 \times 2 \]
Thus,
\[ \sqrt[3]{16} = \sqrt[3]{2^3 \times 2} = \sqrt[3]{2^3} \times \sqrt[3]{2} = 2 \sqrt[3]{2} \]

3. **Simplifying \( \sqrt[3]{54} \)**:
\[ 54 = 27 \times 2 = 3^3 \times 2 \]
Thus,
\[ \sqrt[3]{54} = \sqrt[3]{3^3 \times 2} = \sqrt[3]{3^3} \times \sqrt[3]{2} = 3 \sqrt[3]{2} \]

### Step 2: Substitute back into the expression

Now substituting back into the original expression:
\[ 5\sqrt[3]{250} = 5 \times 5 \sqrt[3]{2} = 25 \sqrt[3]{2} \]
\[ 7\sqrt[3]{16} = 7 \times 2 \sqrt[3]{2} = 14 \sqrt[3]{2} \]
\[ 14\sqrt[3]{54} = 14 \times 3 \sqrt[3]{2} = 42 \sqrt[3]{2} \]

### Step 3: Combine the terms

Now, we can combine all the terms:
\[ 25\sqrt[3]{2} + 14\sqrt[3]{2} - 42\sqrt[3]{2} \]
Combine the coefficients:
\[ (25 + 14 - 42)\sqrt[3]{2} = (-3)\sqrt[3]{2} \]

### Final Answer
Thus, the simplified expression is:
\[ -3\sqrt[3]{2} \]

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