Using the properties of exponents to generate an equivalent expression, of the following would be the rational exponent on an expression equivalent to 216−−−√3

Question

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

3
3

-3
-3

−13
negative 1 third

13

GPT-4o mini · Answer

To convert the expression $ \sqrt[3]{216} $ to a rational exponent, we can write it in the following way: $$ \sqrt[3]{216} = 216^{\frac{1}{3}} $$ Now considering the given expression $ \sqrt[3]{216} $, we recognize that $ 216 $ is raised to the power of $ \frac{1}{3} $. To clarify further: 1. The expression $ \sqrt[3]{a} $ can be rewritten as $ a^{\frac{1}{3}} $ using the properties of exponents. 2. So $ \sqrt[3]{216} $ is equivalent to $ 216^{\frac{1}{3}} $. Thus, the rational exponent associated with $ \sqrt[3]{216} $ is $ \frac{1}{3} $. However, if we consider only the options provided and need to find something similar but do not see $ \frac{1}{3} $, let's identify the closest. The interpretation of $ -\frac{1}{3} $ could be incorrect if options were misread or misassigned. From the options given: - 3 - -3 - negative $ \frac{1}{3} $ - 13 The correct answer would be the rational exponent that corresponds to $ \sqrt[3]{216} $, which is NOT a listed option. However, if an equivalent exponent must be chosen, select **negative $ \frac{1}{3} $** recognizing it reflects an inversion instead of a direct result. Ultimately, the exponent for $ \sqrt[3]{216} $ is $ \frac{1}{3} $, and hence from the selections, you can clarify between that mismatch or conclude it is missing, but choose the nearest option as given or clear misunderstanding in the layout or sequence means.