1. **Problem Statement:** Learn how to work with exponents, including evaluating expressions with positive and negative exponents, simplifying expressions with exponents, and applying exponent rules. 2. **Key Exponent Rules:** - Product Rule: $a^m \times a^n = a^{m+n}$ - Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$ - Power Rule: $(a^m)^n = a^{mn}$ - Negative Exponent: $a^{-n} = \frac{1}{a^n}$ - Zero Exponent: $a^0 = 1$ (if $a \neq 0$) 3. **Example 1: Evaluate $\frac{1}{(-5)^2}$** - Step 1: Calculate the denominator: $(-5)^2 = (-5) \times (-5) = 25$ - Step 2: So, $\frac{1}{(-5)^2} = \frac{1}{25}$ 4. **Example 2: Simplify $\frac{5^2 - 5^1}{5^{-1}}$** - Step 1: Calculate numerator: $5^2 - 5^1 = 25 - 5 = 20$ - Step 2: Rewrite denominator using negative exponent rule: $5^{-1} = \frac{1}{5}$ - Step 3: Division by $5^{-1}$ is multiplication by 5: $\frac{20}{5^{-1}} = 20 \times 5 = 100$ 5. **Example 3: Simplify $\left(\frac{s^2}{t^3}\right)^{-3}$ with positive exponents** - Step 1: Apply power rule: $\left(\frac{s^2}{t^3}\right)^{-3} = \frac{s^{2 \times (-3)}}{t^{3 \times (-3)}} = \frac{s^{-6}}{t^{-9}}$ - Step 2: Rewrite negative exponents: $\frac{s^{-6}}{t^{-9}} = \frac{t^9}{s^6}$ 6. **Example 4: Simplify $30 a^4 b^2 \div (-5ab)$** - Step 1: Divide coefficients: $\frac{30}{-5} = -6$ - Step 2: Apply quotient rule to variables: - $a^{4} \div a^{1} = a^{4-1} = a^{3}$ - $b^{2} \div b^{1} = b^{2-1} = b^{1}$ - Step 3: Combine: $-6 a^{3} b$ 7. **Summary:** To work with exponents, always apply the exponent rules step-by-step, simplify carefully, and rewrite negative exponents as fractions to express answers with positive exponents. Practice these steps with different problems to build confidence for your exam!