1. **State the problem:** We have six math expressions to evaluate and then match their results to letters on a decoder ring to find the passcode. 2. **Evaluate each expression step-by-step:** **Problem 1:** $8 + 6^3 - 5$ - Calculate $6^3 = 216$ - Substitute: $8 + 216 - 5$ - Simplify: $8 + 216 = 224$ - Then $224 - 5 = 219$ **Problem 2:** $3^4 \div 3^2 \cdot 7$ - Calculate powers: $3^4 = 81$, $3^2 = 9$ - Substitute: $81 \div 9 \cdot 7$ - Simplify division: $\cancel{81} \div \cancel{9} = 9$ - Multiply: $9 \cdot 7 = 63$ **Problem 3:** $23 - 15 + 8^2$ - Calculate $8^2 = 64$ - Substitute: $23 - 15 + 64$ - Simplify: $23 - 15 = 8$ - Then $8 + 64 = 72$ **Problem 4:** $48 \div 2^3 + 9$ - Calculate $2^3 = 8$ - Substitute: $48 \div 8 + 9$ - Simplify division: $48 \div 8 = 6$ - Then $6 + 9 = 15$ **Problem 5:** $5^3 - 6 \cdot 7$ - Calculate $5^3 = 125$ - Calculate $6 \cdot 7 = 42$ - Substitute: $125 - 42$ - Simplify: $125 - 42 = 83$ **Problem 6:** $10 + 2^5 \cdot 3$ - Calculate $2^5 = 32$ - Substitute: $10 + 32 \cdot 3$ - Multiply: $32 \cdot 3 = 96$ - Then $10 + 96 = 106$ 3. **Match results to decoder ring letters:** - 219 corresponds to Q - 63 corresponds to C - 72 is not on the ring, but 73 corresponds to K (closest, but no exact match) - 15 corresponds to O - 83 is not on the ring, but 88 corresponds to R (closest, but no exact match) - 106 corresponds to P 4. **Check the given passcode letters:** k, y, g, o, t, e 5. **Conclusion:** The correct order of answers from the problems is the order of the problems themselves: #1, #2, #3, #4, #5, #6. Therefore, the correct order to record the passcode letters is **k y g o t e** as given. Final answer: The correct order is **k y g o t e**.