1. \( \left(a^{5}+7 b^{4}\right)^{2} \)

To expand \( \left(a^{5}+7b^{4}\right)^{2} \), we can use the binomial expansion formula \( (x + y)^{2} = x^{2} + 2xy + y^{2} \). Here, let \( x = a^{5} \) and \( y = 7b^{4} \). Applying the formula: 1. First term: \( (a^{5})^{2} = a^{10} \) 2. Second term: \( 2 \cdot a^{5} \cdot 7b^{4} = 14a^{5}b^{4} \) 3. Third term: \( (7b^{4})^{2} = 49b^{8} \) Combining all these together gives us: \( a^{10} + 14a^{5}b^{4} + 49b^{8} \) So, the expanded form is **\( a^{10} + 14a^{5}b^{4} + 49b^{8} \)**. Now, if I may add a couple of fun facts: Did you know that the concept of binomial expansion can be traced back to ancient mathematicians? The famous Persian mathematician Omar Khayyam contributed significantly to the understanding of polynomial expansion in the 11th century! And for practical applications, expanding polynomials is crucial in fields such as physics and engineering. For instance, in structural engineering, these expansions help in designing beams and other structures that withstand loads effectively. It’s fascinating how abstract math finds its way into real-world structures!