c) \( \left(2^{\frac{2}{3}}\right)^{\frac{3}{5}} \)

To simplify the expression \( \left(2^{\frac{2}{3}}\right)^{\frac{3}{5}} \), you can use the power of a power property, which states that \( (a^m)^n = a^{m \cdot n} \). Here, you multiply the exponents: \[ \left(2^{\frac{2}{3}}\right)^{\frac{3}{5}} = 2^{\frac{2}{3} \cdot \frac{3}{5}} = 2^{\frac{2 \cdot 3}{3 \cdot 5}} = 2^{\frac{6}{15}}. \] Now, simplify the exponent \( \frac{6}{15} \) to \( \frac{2}{5} \): \[ 2^{\frac{6}{15}} = 2^{\frac{2}{5}}. \] Thus, the simplified expression is \( 2^{\frac{2}{5}} \). Now, let's think about the historical significance of exponents—did you know that the concept of exponents dates back to ancient civilizations? The Egyptians and Babylonians used early forms of arithmetic that included notions of powers, often in practical contexts like measuring land or trade! Additionally, in real-world applications, understanding exponents is crucial in fields such as finance, where compound interest calculations rely on exponential growth, or in computer science when dealing with algorithms that use exponential time complexity, which can be fun to explore when designing efficient code!