1. The problem is to evaluate powers of negative numbers and understand their results. 2. The formula for powers is $a^n = a \times a \times \cdots \times a$ ($n$ times). 3. Important rule: When raising a negative number to an even power, the result is positive because multiplying an even number of negative factors results in a positive product. 4. When raising a negative number to an odd power, the result is negative because multiplying an odd number of negative factors results in a negative product. 5. Let's verify each given expression: - $a = (-3)^2 = (-3) \times (-3) = 9$ - $b = (-3)^3 = (-3) \times (-3) \times (-3) = 9 \times (-3) = -27$ - $c = (-3)^4 = (-3)^2 \times (-3)^2 = 9 \times 9 = 81$ - $d = (-5)^3 = (-5) \times (-5) \times (-5) = 25 \times (-5) = -125$ 6. Each calculation follows the rule about even and odd powers of negative numbers. Final answers: $a = 9$ $b = -27$ $c = 81$ $d = -125$