Question

Which of the following expressions is equivalent to 3--√216/27^1/3 when applying the exponent properties?(1 point)
Responses

18
18

8
8

2
2

1/2

Answer 1

To simplify the expression \( 3 - \sqrt{216}/27^{1/3} \), we can break it down step by step.

1. **Calculate \( 27^{1/3} \)**:
\[
27 = 3^3 \quad \text{so} \quad 27^{1/3} = (3^3)^{1/3} = 3^{3 \cdot \frac{1}{3}} = 3^1 = 3
\]

2. **Calculate \( \sqrt{216} \)**:
We can express \( 216 \) as a product of prime factors:
\[
216 = 2^3 \cdot 3^3
\]
Therefore,
\[
\sqrt{216} = \sqrt{2^3 \cdot 3^3} = \sqrt{2^3} \cdot \sqrt{3^3} = 2^{3/2} \cdot 3^{3/2} = 2\sqrt{2} \cdot 3\sqrt{3} = 6\sqrt{6}
\]

3. **Substitute back** into the expression:
\[
3 - \frac{\sqrt{216}}{27^{1/3}} = 3 - \frac{6\sqrt{6}}{3}
\]
\[
= 3 - 2\sqrt{6}
\]

Since this does not correspond to the provided options (18, 8, 2, 1/2), it appears that we had a miscalculation either in rewriting the expression or possibly in the interpretation of the question.

To address the original expression more directly:

Given:
\[
x - \frac{\sqrt{216}}{27^{1/3}}
\]
Express everything in terms of base \( 3 \):

\(\sqrt{216} = 6\sqrt{6}\) and \(27^{1/3} = 3\). Thus:

\[
= 3 - 2\sqrt{6}
\]

However, we directly see that it creates issues, as no integer or simple fraction emerges. My assumption would be a recheck had you initially provided it correctly. If it's calculated directly as shown; none of the answer choices seem to align.

Therefore, please check the original details to ensure the values presented are correct. There might be a need for elaboration or re-evaluation based on \(-\sqrt{}\) or behavior in the context of roots and fractions not provided directly within integer bounds or direct acceptance.