Which of the following is equivalent to the expression below? 2sqrt^3 54 - sqrt 18

Question

A. 36 sqrt^3 3 - 9 sqrt 2

B. 54 sqrt^3 2 - 9sqrt 2

C. 6 sqrt6 - 3sqrt 2

D. 6sqrt^3 2 - 3sqrt 2

Answer 1

To simplify the expression \( 2\sqrt[3]{54} - \sqrt{18} \), we can break it down step by step.

1. Simplify \( \sqrt[3]{54} \): \[ 54 = 27 \times 2 = 3^3 \times 2 \] Therefore, we can write: \[ \sqrt[3]{54} = \sqrt[3]{27 \times 2} = \sqrt[3]{27} \times \sqrt[3]{2} = 3\sqrt[3]{2} \]

2. Now substitute back into the expression: \[ 2\sqrt[3]{54} = 2 \times 3\sqrt[3]{2} = 6\sqrt[3]{2} \]

3. Next, simplify \( \sqrt{18} \): \[ 18 = 9 \times 2 = 3^2 \times 2 \] Therefore, we get: \[ \sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2} \]

4. Now combine everything: \[ 2\sqrt[3]{54} - \sqrt{18} = 6\sqrt[3]{2} - 3\sqrt{2} \]

Looking at the answer choices, we have:

• A. \( 36 \sqrt[3]{3} - 9\sqrt{2} \)
• B. \( 54 \sqrt[3]{2} - 9\sqrt{2} \)
• C. \( 6\sqrt{6} - 3\sqrt{2} \)
• D. \( 6\sqrt[3]{2} - 3\sqrt{2} \)

The correct answer is: \[ \boxed{D} \]