24. \( \sqrt[3]{x+4}-1=0 \)

To solve the equation \( \sqrt[3]{x+4}-1=0 \), we first isolate the cube root by adding 1 to both sides: \[ \sqrt[3]{x+4} = 1. \] Next, we'll eliminate the cube root by cubing both sides: \[ x + 4 = 1^3, \] which simplifies to: \[ x + 4 = 1. \] Now, subtract 4 from both sides to solve for \( x \): \[ x = 1 - 4, \] resulting in: \[ x = -3. \] So, the solution to the equation is \( x = -3 \). It's always a good idea to check your solution. If you substitute \( -3 \) back into the original equation: \[ \sqrt[3]{-3+4}-1 = \sqrt[3]{1}-1 = 1-1 = 0. \] Since this holds true, our solution \( x = -3 \) is confirmed! If you're curious about cube roots, they were majorly studied in ancient civilizations. The Babylonians, as early as 2000 BC, employed methods for solving cubic equations, which paved the way for more complex mathematics. Their innovative approaches to approximating cube roots were impressive for such an early era! For real-world application, cube roots can appear in various fields including engineering and architecture, especially when dealing with volumes. For example, if you're calculating the dimensions of a cube that holds a specific volume, you would take the cube root of that volume to find the length of each side. So, whether you're building a birdhouse or designing a skyscraper, those cube roots are more important than you might think!