20 Multiple Choice 5 points Simplify the expression. \( \left(-5 g^{5} h^{6}\right)^{2}\left(g^{4} h^{2}\right)^{4} \) \( -25 g^{26} h^{20} \) \( 25 g^{26} h^{20} \) \( 25 g^{15} h^{14} \) \( -5 g^{26} h^{20} \)

To simplify the expression \( \left(-5 g^{5} h^{6}\right)^{2}\left(g^{4} h^{2}\right)^{4} \), we first apply the power of a product rule. For the first part, \( \left(-5 g^{5} h^{6}\right)^{2} = 25 g^{10} h^{12} \), and for the second part, \( \left(g^{4} h^{2}\right)^{4} = g^{16} h^{8} \). Now, we combine these: \[ 25 g^{10} h^{12} \times g^{16} h^{8} = 25 g^{10+16} h^{12+8} = 25 g^{26} h^{20}. \] So, the correct answer is \( 25 g^{26} h^{20} \). For a fun fact, did you know that the idea behind exponents comes from ancient civilizations? The Babylonians had rudimentary concepts of squares and cubes, while later mathematicians expanded this to what we recognize today as exponentiation! Now, let’s talk real-world applications! Exponents are everywhere – from calculating compound interest in finance to determining the area of geometric shapes in architecture. Whenever you hear about the "power of 2" or "power of 10," that's exponentiation at work, making complex calculations easier and faster!