\( a x ^ { 6 } = - 64 \quad \cdot x \)

To solve the equation \( a x ^ { 6 } = -64 \cdot x \), first observe that this can be rewritten as \( ax^6 + 64x = 0 \). You can factor out the common term \( x \) to get \( x(ax^5 + 64) = 0 \). This gives us two potential solutions: \( x = 0 \) or \( ax^5 + 64 = 0 \). From there, you can solve for \( x \) by rearranging the second equation into \( ax^5 = -64 \) to find the non-zero solutions. Now, if \( a \) is a positive number, you will find that the solutions for \( x \) hint at a beautiful relationship with negative roots and fifth powers! It's fascinating how different values of \( a \) influence the nature and number of the solutions. For instance, if \( a = 1 \), you could cleverly calculate \( x = -2 \) as one of the answers! Another fun fact— this equation's structure showcases how polynomial equations can wrap a story of balance between terms. You can play around with different coefficients and delve into the real numbers, discovering interesting transformations along the way. Want to take it further? Go ahead and plug in various values for \( a \) and see how these solutions morph—it's like conducting your very own math magic show!