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)

Question

Responses

3
3

−3
negative 3

−13
negative Start Fraction 1 over 3 End Fraction

13

Answer 1

To express \( 216^{-\frac{1}{3}} \) using properties of exponents, we need to understand what the expression represents.

1. The expression \( \sqrt[3]{216} \) means the cube root of 216.
2. The negative exponent indicates that we take the reciprocal.

The full expression can be interpreted as:

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

The cube root of 216 is:

\[ \sqrt[3]{216} = 6 \]

Thus, we can express our original expression as:

\[ 216^{-\frac{1}{3}} = \frac{1}{6} \]

Now, in simplified terms, the rational exponent on the original expression \( 216^{-\frac{1}{3}} \) is \( -\frac{1}{3} \).

So, the answer is:

−13 (negative Start Fraction 1 over 3 End Fraction).