To rewrite the expression \( 216^{\frac{1}{3}} \), we can start by recognizing that a radical expression can be rewritten using rational exponents. The expression \( 216^{\frac{1}{3}} \) means that we take the cube root of 216.
The properties of exponents tell us that \( a^{\frac{m}{n}} \) is equivalent to \( \sqrt[n]{a^m} \). In this case, taking the cube root corresponds to raising something to the power of \(\frac{1}{3}\).
Therefore, \( 216^{\frac{1}{3}} \) is the same as \( \sqrt[3]{216} \), which simplifies to 6 since \( 6^3 = 216 \).
None of the options provided (13, \( \frac{1}{3} \), -13, -\( \frac{1}{3} \), 3, -3) represents an equivalent expression for \( 216^{\frac{1}{3}} \).
If the expression actually given was \( 216^{-\frac{1}{3}} \), then it would mean the cube root of \( 216 \) raised to a negative power, which would give us \( \frac{1}{216^{\frac{1}{3}}} = \frac{1}{6} \).
However, considering just \( 216^{\frac{1}{3}} \), the equivalent rational exponent is indeed \( \frac{1}{3} \).
Thus, the correct response to your question should be: \(\frac{1}{3}\).
1 answer