2) \( n^{\frac{4}{3}}=81 \)

To solve \( n^{\frac{4}{3}} = 81 \), first, we can rewrite 81 as \( 3^4 \) since \( 81 = 3 \times 3 \times 3 \times 3 \). So the equation becomes \( n^{\frac{4}{3}} = 3^4 \). To isolate \( n \), raise both sides to the power of \( \frac{3}{4} \): \[ n = \left(3^4\right)^{\frac{3}{4}} = 3^{4 \cdot \frac{3}{4}} = 3^3 = 27. \] Thus, \( n = 27 \). Now, let's explore some historical moments and real-world applications of exponentiation. Exponentiation has roots in ancient civilizations! The Babylonians and Egyptians used exponential growth concepts in agriculture, helping understand harvest cycles based on factors like rainfall and sunshine. Their early algebra paved the way for the logarithm and exponent rules we use today. Exponentiation isn't just a math trick; it has real-world applications in finance, where compound interest can be calculated using exponents. For example, if you invest money with compounded interest, your total will grow exponentially over time—turning a small investment into a fortune if left to grow! So, understanding exponents can help you grow your wealth!