Octal to decimal conversion practice: Convert the octal number 3137₈ to its decimal (base 10) value.

Introduction / Context:Understanding place values in different bases allows you to convert any positional number system into decimal. Octal digits are weighted by powers of 8 from right to left: 8^0, 8^1, 8^2, and so on.

Given Data / Assumptions:

• Given octal: 3 1 3 7 (i.e., 3137₈).
• Digits are valid octal digits (0–7).
• Goal: compute the exact decimal value.

Concept / Approach:Use the positional expansion: For digits dₙ…d₂ d₁ d₀ in base 8, value = Σ dᵢ * 8^i. Multiply each digit by the corresponding power of 8 and sum the results to obtain the base-10 number.

Step-by-Step Solution:Digits and weights: 38^3 + 18^2 + 38^1 + 78^0.Compute powers: 8^3 = 512, 8^2 = 64, 8^1 = 8, 8^0 = 1.Multiply: 3512 = 1536; 164 = 64; 38 = 24; 71 = 7.Sum: 1536 + 64 + 24 + 7 = 1631.Therefore, 3137₈ = 1631₁₀.

Verification / Alternative check:Quick sanity: Highest place 3*512 = 1536 already, so the total must be a bit above 1536; 1631 is consistent. Calculator check will match 1631.

Why Other Options Are Wrong:1632₁₀, 1531₁₀, and 1931₁₀ result from digit or power mistakes. Only 1631₁₀ is correct.

Common Pitfalls:Reversing digit order; using 10^i instead of 8^i; arithmetic slips when summing partial results.

Final Answer:163110