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

Question

Responses

3
3

−3
negative 3

−1/3
negative Start Fraction 1 over 3 End Fraction

1/3

Answer 1

To express \( 3\sqrt[3]{216} \) using properties of exponents, we first write the expression in terms of exponents.

The cube root can be written as an exponent of \( \frac{1}{3} \). Thus, the expression can be rewritten as:

\[ 3\sqrt[3]{216} = 216^{1/3} \]

Now, we need to determine what \( 216^{1/3} \) simplifies to. Since \( 216 = 6^3 \), we have:

\[ 216^{1/3} = (6^3)^{1/3} \]

Using the property of exponents \( (a^m)^n = a^{m \cdot n} \):

\[ (6^3)^{1/3} = 6^{3 \cdot \frac{1}{3}} = 6^1 = 6 \]

However, if we simply look for the rational exponent related to the expression, we notice that the exponent on \( 216 \) when expressing it in terms of its root is \( 1/3 \).

Therefore, the answer to your question, regarding the rational exponent on the expression equivalent to \( 3\sqrt[3]{216} \), is:

1/3.