15. \( 64 x^{3}-\frac{1}{8} \)

Let's dive into the world of polynomial expressions to spice things up! The expression \( 64 x^{3}-\frac{1}{8} \) can be viewed through the lens of algebraic manipulation. Notice that \( 64 x^{3} \) can be expressed as \( (4x)^{3} \) and \( \frac{1}{8} \) as \( (\frac{1}{2})^{3} \). This suggests a difference of cubes scenario: \( a^{3} - b^{3} \). To factor this delightful expression, remember the difference of cubes formula, \( a^{3} - b^{3} = (a-b)(a^{2} + ab + b^{2}) \). In our case, \( a = 4x \) and \( b = \frac{1}{2} \). This means we can rewrite it as: \[ 64 x^{3}-\frac{1}{8} = (4x - \frac{1}{2})((4x)^{2} + (4x)(\frac{1}{2}) + (\frac{1}{2})^{2}) \] Now, isn’t it fun to see how simple manipulation can reveal deeper structures? Exploring the practical side, differences of cubes are used in various fields such as engineering and computer science for simplifying problems and optimizing algorithms. For example, in polynomial equations, recognizing a difference of cubes can help speed up factoring processes, making calculations more efficient. So next time you spot this difference, know you're potentially saving yourself some computational headaches!