What is $3^{\frac{2}{3}}$ equal to?
$\sqrt[3]{9}$
$2 \sqrt{9}$
$\sqrt[3]{27}$
$2 \sqrt{27}$
Answer (1)
Rewrite the expression using the property a n m = n a m .
Apply this to 3 3 2 to get 3 3 2 .
Simplify the expression to 3 9 .
The equivalent expression is 3 9 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to 3 3 2 from the given options. The options are 3 9 , 2 9 , 3 27 , 2 27 .
Applying Properties of Exponents and Radicals We need to rewrite 3 3 2 using the properties of exponents and radicals. Specifically, we will use the property a n m = n a m .
Rewriting the Expression Applying the property to 3 3 2 , we get 3 3 2 = 3 9 .
Finding the Equivalent Expression Comparing the result with the given options, we find that 3 9 is one of the options.
Conclusion Therefore, 3 3 2 is equal to 3 9 .
Examples
Understanding fractional exponents is crucial in various fields, such as physics and engineering, where they are used to model phenomena like the growth of crystals or the decay of radioactive materials. For instance, if you're analyzing the rate at which a certain material dissolves in a solution, the rate might be proportional to a fractional power of the concentration of the solute. This problem demonstrates how to convert between exponential and radical forms, which is a fundamental skill in these applications.
