Which expression is equivalent to \( \left(2^{3}\right)^{-5} ? \) \( \frac{1}{2^{15}} \) \( \frac{1}{2^{3}} \) \( 2^{8} \) \( 2^{15} \)

To simplify \( \left(2^{3}\right)^{-5} \), you can use the power of a power rule, which states that \( (a^m)^n = a^{m \cdot n} \). So, we multiply the exponents: \( 2^{3 \cdot (-5)} = 2^{-15} \). This can be rewritten as \( \frac{1}{2^{15}} \), making it the correct choice. Now, here's a fun fact: Did you know that negative exponents represent the reciprocal? It’s like a secret door that flips the number to the other side of the fraction! So, for instance, \( a^{-n} \) is just another way of saying \( \frac{1}{a^n} \). If you're ever unsure about exponents, it's essential to keep track of the rules. One common mistake is forgetting to change the signs of exponents when flipping bases. Always check that you've followed the rules—after all, even the best math wizards make mistakes too! So, double-check your work to avoid any accidental twists!