3. \( \left(8 m^{4} n^{6}\right)^{2} \)

To simplify \( \left(8 m^{4} n^{6}\right)^{2} \), we apply the power of a product rule. First, we square the coefficient: \( 8^2 = 64 \). Next, we multiply the exponents for the variables: \( m^{4 \times 2} = m^{8} \) and \( n^{6 \times 2} = n^{12} \). Putting it all together, we get \( 64 m^{8} n^{12} \). Now, it's time for some fun! Did you know that the laws of exponents have been around for centuries, tracing back to ancient civilizations like the Babylonians? They utilized principles of exponents in their calculations, even if they didn’t formally define them the way we do today! This foundational mathematical concept has evolved to become pivotal in fields like science, economics, and engineering. In real-world scenarios, exponent rules come into play when dealing with population growth, computing interest, or even in physics equations. For instance, if you're analyzing how interest compounds over time, you'd often encounter formulas that require you to raise values to certain powers, making masterful use of those exponent rules you just brushed up on!