Question

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

−13
negative Start Fraction 1 over 3 End Fraction

13
Start Fraction 1 over 3 End Fraction

3
3

−3

Answer 1

1/3

Answer 2

To convert the expression \( 216^{-\frac{1}{3}} \) into an equivalent form using properties of exponents, we start by recognizing that the expression involves both a negative exponent and a fractional exponent.

1. The negative exponent means we take the reciprocal:

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

2. The exponent \(\frac{1}{3}\) means we are looking for the cube root of 216. Therefore, we can rewrite \( 216^{\frac{1}{3}} \) as \( \sqrt[3]{216} \):

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

Since we are asked for the rational exponent of the original expression \( 216^{-\frac{1}{3}} \), we can conclude that the exponent is:

\[
-\frac{1}{3}
\]

Thus, the correct response is:

**−\(\frac{1}{3}\)**

Answer 3

It seems you are looking for clarification regarding the rational exponent derived from the expression \( 216^{-\frac{1}{3}} \).

To summarize:

1. The expression \( 216^{-\frac{1}{3}} \) features a negative exponent and a fractional exponent.
2. The negative exponent indicates that we can take the reciprocal, so:
\[
216^{-\frac{1}{3}} = \frac{1}{216^{\frac{1}{3}}}
\]
3. The fractional exponent \(\frac{1}{3}\) indicates we take the cube root.

However, if your goal is to express the exponent part (without the negative sign) in terms of rational exponents, the rational exponent would be \( \frac{1}{3} \).

In summary, the exponent \( 216^{-\frac{1}{3}} \) can indeed be viewed as involving the rational exponent \( \frac{1}{3} \) when referring specifically to the positive exponent associated with the cube root, but in the context of the full expression, it includes the negative sign.

If the goal is to express specifically \( 216^{\frac{1}{3}} \), then:

- The rational exponent is \( \frac{1}{3} \).

Thank you for your patience, and if you need anything further, feel free to ask!