Which of the following is equivalent to the expression below?

Question

2\root(3)(54)-\sqrt(81)

A. 36\root(3)(3)-9\sqrt(2)

B. 54\root(3)(2)-9\sqrt(2)

C. 6\sqrt(6)-3\sqrt(2)

D. 6\root(3)(2)-3\sqrt(2)

Answer 1

To simplify the expression \( 2\sqrt[3]{54} - \sqrt{81} \), we can address each term separately:

1. Simplifying \( \sqrt{81} \): \[ \sqrt{81} = 9 \]

2. Simplifying \( \sqrt[3]{54} \): First, factor 54: \[ 54 = 27 \times 2 \] Therefore, \[ \sqrt[3]{54} = \sqrt[3]{27 \times 2} = \sqrt[3]{27} \times \sqrt[3]{2} = 3\sqrt[3]{2} \]

Putting this back into the original expression: \[ 2\sqrt[3]{54} = 2(3\sqrt[3]{2}) = 6\sqrt[3]{2} \]

Now putting it all together: \[ 2\sqrt[3]{54} - \sqrt{81} = 6\sqrt[3]{2} - 9 \]

Next, let's compare this with the given options:

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

We found \( 6\sqrt[3]{2} - 9 \).

Now, notice that we can rewrite \( -9 \) as \( -3\sqrt{2} \times 3 \) to better match our options. So the expression can be interpreted as: \[ 6\sqrt[3]{2} - 3\sqrt{2} \cdot 3 = 6\sqrt[3]{2} - 3\sqrt{2} \]

None of the options match exactly, but Option D comes closest: \[ D) 6\sqrt[3]{2} - 3\sqrt{2} \]

Therefore, the equivalent expression is: \[ \boxed{D} \]