Using the properties of exponents to generate an equivalent expression, which of the following would be the rational exponent on an expression equivalent to 3−−−√216 ?(1 point)

Question

Responses

3
3

−3
negative 3

−13
negative Start Fraction 1 over 3 End Fraction

13

Answer 1

To rewrite the expression \( \sqrt[3]{216} \) using rational exponents, we need to express the cube root in exponent form. The cube root can be written as raising to the power of \( \frac{1}{3} \).

So, we have:

\[ \sqrt[3]{216} = 216^{\frac{1}{3}} \]

Now, we can also express \( 216 \) in terms of its prime factorization:

\[ 216 = 6^3 \]

Thus,

\[ 216^{\frac{1}{3}} = (6^3)^{\frac{1}{3}} = 6^{3 \cdot \frac{1}{3}} = 6^1 = 6 \]

Therefore, the rational exponent that corresponds to the expression \( \sqrt[3]{216} \) is \( \frac{1}{3} \) in the exponent form.

However, it seems that you are looking for an option among the responses provided, especially since none of the answers correctly reflect this understanding. Based on the original expression \( 3^{-\sqrt{216}} \), the exponent will depend on what you intended to express. Assuming we keep just the \( \sqrt[3]{216} \).

But for clarity, if you meant the original \( \sqrt[3]{216} \), then the answer does not exist in this selection as they are based on another interpretation.

The closest option to match a rational interpretation of \( 216^{\frac{1}{3}} \) would lead us to:

So the answer is: \[ \text{None of the provided options match the correct rational exponent.} \]

If needed, can clarify or further breakdown any other aspect based on the answers given.