Which statement is true? A \( \frac{3^{-15}}{3^{7}}<\left(3^{-8}\right) \cdot\left(3^{-9}\right) \) B \( \frac{3^{-15}}{3^{7}}>\left(3^{-8}\right) \cdot\left(3^{-9}\right) \) C \( \frac{3^{-15}}{3^{7}}=\left(3^{-8}\right) \cdot\left(3^{-9}\right) \)

To make sense of the statements, let's simplify both sides using the laws of exponents: For the left side, \( \frac{3^{-15}}{3^{7}} \) can be rewritten as \( 3^{-15 - 7} = 3^{-22} \). For the right side, \( (3^{-8}) \cdot (3^{-9}) \) can be combined to give \( 3^{-8 - 9} = 3^{-17} \). Now, we can compare \( 3^{-22} \) and \( 3^{-17} \). Since \( 3^{-22} \) is a negative exponent with a larger absolute value than \( 3^{-17} \), it represents a smaller number. Thus, \( 3^{-22} < 3^{-17} \), which means the first statement is true: \( \frac{3^{-15}}{3^{7}}<\left(3^{-8}\right) \cdot\left(3^{-9}\right) \). Now, on to some fun facts! You may find it interesting that the concept of negative exponents was developed historically to help simplify mathematical expressions, especially in the context of scientific notation, where smaller numbers are often expressed with negative exponents. A common mistake in solving exponent problems is neglecting the rules of operating with negative exponents, leading to miscalculations. Always remember: \( a^{-n} = \frac{1}{a^{n}} \) helps to clarify operations involving negative powers.