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

−13
negative Start Fraction 1 over 3 End Fraction

3
3

−3
negative 3

13

Answer 1

To find the rational exponent of the expression \( 216^{-\frac{1}{3}} \), we can rewrite the expression using the properties of exponents.

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

• The negative exponent indicates that we are taking the reciprocal of the base raised to the positive exponent.
• The fractional exponent indicates the root; specifically, the denominator of the fraction represents the root.

Thus:

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

Now, to determine the form of \( 216^{\frac{1}{3}} \):

• The cube root of \( 216 \) is \( 6 \) since \( 6^3 = 216 \).

Therefore:

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

From the options given, the equivalent expression with the rational exponent in \( 216^{-1/3} \) is simply:

\(-\frac{1}{3}\)

So the rational exponent on an expression equivalent to \( 216^{-\frac{1}{3}} \) is:

\(-\frac{1}{3}\)

Thus, the answer is:

−1/3