\( ( \frac { a } { b } ) ^ { - n } \)

Certainly! Let's analyze and simplify the expression you've provided: \[ \left( \frac{a}{b} \right)^{-n} \] ### Understanding Negative Exponents A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. In general: \[ x^{-n} = \frac{1}{x^n} \] ### Applying to Your Expression Given your expression: \[ \left( \frac{a}{b} \right)^{-n} \] 1. **Take the Reciprocal:** Applying the negative exponent rule: \[ \left( \frac{a}{b} \right)^{-n} = \left( \frac{b}{a} \right)^n \] 2. **Final Simplified Form:** \[ \left( \frac{b}{a} \right)^n \] ### Alternative Interpretation If you prefer to express it as a single fraction raised to the positive exponent, it's already in its simplest form. However, expressing it as shown emphasizes the effect of the negative exponent by flipping the fraction. ### Example Let's see a numerical example to illustrate this: Suppose \( a = 2 \), \( b = 3 \), and \( n = 4 \). 1. **Original Expression:** \[ \left( \frac{2}{3} \right)^{-4} \] 2. **Apply Negative Exponent Rule:** \[ \left( \frac{2}{3} \right)^{-4} = \left( \frac{3}{2} \right)^4 \] 3. **Calculate the Power:** \[ \left( \frac{3}{2} \right)^4 = \frac{3^4}{2^4} = \frac{81}{16} \] So, \[ \left( \frac{2}{3} \right)^{-4} = \frac{81}{16} \] ### Summary - **Original Expression:** \( \left( \frac{a}{b} \right)^{-n} \) - **Simplified Form:** \( \left( \frac{b}{a} \right)^n \) This manipulation is particularly useful in algebra to simplify expressions, solve equations, or when dealing with exponential and logarithmic functions. If you have any specific questions or need further clarification on how to use this in a particular context, feel free to ask!