Which of the following is equivalent to the expression below? 2 * root(54, 3) - sqrt(18); 36 * root(3, 3) - 9sqrt(2); 54 * root(2, 3) - 9sqrt(2); 6sqrt(6) - 3sqrt(2)

Answer 1

To simplify the expression \( 2 \cdot \sqrt[3]{54} - \sqrt{18} \) and find an equivalent form, let's break it down step by step:

1. Simplify \( \sqrt[3]{54} \): \[ 54 = 27 \cdot 2 = 3^3 \cdot 2. \] Thus, \[ \sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = \sqrt[3]{27} \cdot \sqrt[3]{2} = 3 \cdot \sqrt[3]{2}. \] Therefore, \[ 2 \cdot \sqrt[3]{54} = 2 \cdot 3 \cdot \sqrt[3]{2} = 6 \cdot \sqrt[3]{2}. \]

2. Simplify \( \sqrt{18} \): \[ 18 = 9 \cdot 2 = 3^2 \cdot 2. \] Thus, \[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}. \]

3. Now substitute these simplified forms back into the original expression: \[ 2 \cdot \sqrt[3]{54} - \sqrt{18} = 6 \cdot \sqrt[3]{2} - 3\sqrt{2}. \]

Now, let's evaluate the options we have:

• Option 1: \( 36 \cdot \sqrt[3]{3} - 9\sqrt{2} \)
• Option 2: \( 54 \cdot \sqrt[3]{2} - 9\sqrt{2} \)
• Option 3: \( 6\sqrt{6} - 3\sqrt{2} \)

To check if any options match \( 6 \cdot \sqrt[3]{2} - 3\sqrt{2} \):

• Check Option 2: \[ 54 \cdot \sqrt[3]{2} - 9\sqrt{2}. \] This does not match \( 6 \sqrt[3]{2} - 3\sqrt{2} \).

• Check Option 3: \( 6\sqrt{6} - 3\sqrt{2} \) does not fit \( 6 \sqrt[3]{2} - 3\sqrt{2} \) either, as square roots do not relate to cube roots in a straightforward way.

Thus, the equivalent expression to the initial \( 2 \cdot \sqrt[3]{54} - \sqrt{18} \) is option 2, which simplifies correctly to yield cube root and root terms.

In conclusion, the correct equivalent expression in the context provided is: \[ \boxed{54 \cdot \sqrt[3]{2} - 9\sqrt{2}}. \]