Solved Examples: Evaluating Expressions with Negative Exponents

📚 Quick Study Guide

🔍 Definition: A negative exponent indicates the reciprocal of the base raised to the positive of that exponent. $a^{-n} = \frac{1}{a^n}$ (where $a \neq 0$). 💡 Rule 1: To eliminate a negative exponent, move the base and its exponent to the opposite side of the fraction bar. If it's in the numerator, move it to the denominator, and vice versa. 📝 Rule 2: When simplifying expressions with negative exponents, remember to apply the order of operations (PEMDAS/BODMAS). ➗ Rule 3: Any non-zero number raised to the power of zero is equal to 1. $a^0 = 1$ (where $a \neq 0$). ➕ Example 1: Simplify $2^{-3}$. Solution: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$. ➖ Example 2: Simplify $\frac{1}{5^{-2}}$. Solution: $\frac{1}{5^{-2}} = 5^2 = 25$. ➗ Example 3: Simplify $(3x)^{-2}$. Solution: $(3x)^{-2} = \frac{1}{(3x)^2} = \frac{1}{9x^2}$.

Practice Quiz

1. What is the simplified form of $4^{-2}$?
   1. $\frac{1}{16}$
   2. 16
   3. -8
   4. -16
2. Simplify: $x^{-5}$?
   1. $5x$
   2. $\frac{1}{x^5}$
   3. $-x^5$
   4. $x^5$
3. Simplify: $\frac{1}{3^{-4}}$?
   1. $\frac{1}{81}$
   2. -81
   3. 81
   4. -$\frac{1}{81}$
4. What is the value of $(-2)^{-3}$?
   1. 8
   2. -8
   3. $\frac{1}{8}$
   4. $\frac{-1}{8}$
5. Simplify: $(2a)^{-3}$?
   1. $8a^3$
   2. $\frac{1}{8a^3}$
   3. $\frac{1}{2a^3}$
   4. $-8a^3$
6. Simplify: $5^0 + 5^{-1}$?
   1. 0
   2. $\frac{1}{5}$
   3. $\frac{6}{5}$
   4. 6
7. What is the simplified form of $\frac{x^{-2}}{y^{-3}}$?
   1. $\frac{y^3}{x^2}$
   2. $\frac{x^2}{y^3}$
   3. $x^2y^3$
   4. $\frac{1}{x^2y^3}$