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).