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

−3
negative 3

13
Start Fraction 1 over 3 End Fraction

−13
negative Start Fraction 1 over 3 End Fraction

3

Answer 1

To express \( 216^{-\frac{1}{3}} \) using the properties of exponents, we start by rewriting the square root and the exponent.

The expression \( 216^{-\frac{1}{3}} \) can be interpreted as follows:

1. The term \( 216^{\frac{1}{3}} \) denotes the cube root of 216. Since we're taking the cube root and then applying a negative exponent, the negative indicates a reciprocal.

Given that:

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

Therefore, the value of \( 216^{-\frac{1}{3}} \) is \( 216 \) raised to the power of \(-1/3\).

Hence, the rational exponent on the equivalent expression would be:

**−13** (negative \( \frac{1}{3} \)).

So the correct response listed is **−13** (negative \( \frac{1}{3} \)).