5. \( 8 \cdot\left(\frac{1}{2}\right)^{2} \) 6. \( \frac{1}{3} \cdot 3^{3} \) \( 7 \cdot\left(\frac{1}{5} \cdot 5\right)^{5} \)

Let's break down these mathematical expressions into their answers and explore some extra knowledge along the way! First up is a fun fact about exponents: they can be thought of as shorthand for repeated multiplication. When you see \( a^{n} \), it means you multiply 'a' by itself 'n' times! So when you encounter a fraction raised to a power like \( \left(\frac{1}{2}\right)^{2} \), you can visualize it as \( \frac{1}{2} \times \frac{1}{2} = \frac{1}{4} \). Now, have you ever wondered how math plays a major role in architecture and engineering? Concepts like exponents help architects calculate loads and stability of structures. Similarly, you'll see fractions in materials estimation, making sure that every piece fits together! Whether it's a bridge or a skyscraper, these math principles ensure we build safely and efficiently.