Simplify the following expression, working over
𝑍
13
:

3
−
1
2
10
=

Question

Simplify the following expression, working over 
𝑍
13
:

3
−
1
2
10
=

BlackTom AI · Accepted Answer

The question presents: 'Simplify the following expression, working over Z13: 3 − 1/2 10 =', and the answer field shows a single value '10' but the answer options list appears to be empty. Because the answer choices are not provided, we cannot map a final numeric result to a labeled option. We can still explore how to interpret and simplify the given expression under the standard modular arithmetic setup over Z13.

First, restating the problem in a concrete way: we are to compute a value in the field Z13, given an expression that looks like 3 minus something involving 1/2 and 10. A natural interpretation in modular arithmetic is that 1/2 denotes the multiplicative inverse of 2 modulo 13, since 2 has an inverse in Z13.

Interpretation A: 3 − (1/2) × 10 in Z13
- Find the inverse of 2 modulo 13: 2 × 7 = 14 ≡ 1 (mod 13), so 1/2 ≡ 7 (mod 13).
- Multiply this inverse by 10: 7 × 10 = 70 ≡ 70 − 65 = 5 (mod 13).
- Subtract from 3: 3 − 5 ≡ −2 ≡ 11 (mod 13).
- Therefore, under this interpretation, the result would be 11 in Z13.

Interpretation B: 3 − (1/2^10) or 3 − (1/2)^{10}
- If the expression meant 3 − (1/2)^{10}, we would compute 1/2 ≡ 7, then 7^{10} mod 13 and subtract from 3. Since 7^2 ≡ 49 ≡ 10 (mod 13), and powers of 7 cycle with period tied to the group order, this becomes a longer calculation, but still well-defined in Z13.
- If the expression meant 3 − 1/(2 × 10) (i.e., 3 − (1/(2·10))), you would first compute 2·10 ≡ 20 ≡ 7 (mod 13), then find the inverse of 7 modulo 13 (which is 2, since 7×2 = 14 ≡ 1), giving 3 − 2 ≡ 1 (mod 13).
- These alternative readings produce different results, illustrating that decoding the exact intended expression is crucial for a correct simplification.

Interpretation C: 3 − 1/2 10 with a missing operator between 1/2 and 10
- If the expression is 3 − (1/2) 10 with an implied multiplication, we return to Interpretation A and obtain 11.
- If the intended meaning was 3 − (1)/(2×10), that would be 3 − 1/20, but 20 ≡ 7 (mod 13), so 1/20 ≡ 1/7 ≡ 2, giving 3 − 2 ≡ 1 (mod 13).
- If the expression was 3 − (1/2) + 10 or 3 − 1/(2,10) with a comma, different parse yields different results; modular expressions depend critically on the exact parsing and the placement of parentheses.

Notes on missing answer options:
- Since the provided answer options field is empty, there is no explicit choice to verify against. In typical multiple-choice tasks, a concrete option would map to the computed residue in Z13.
- If you have access to the original formatting (for example, whether there were parentheses or a specific operator between the 1/2 and the 10), returning to that precise structure is essential to settle on the correct residue.

Summary of key steps you would perform given a clear interpretation (A):
- Identify the field: work modulo 13.
- Determine the inverse of 2 in Z13, which is 7.
- Compute the product with 10: 7 × 10 ≡ 5 (mod 13).
- Subtract from 3: 3 − 5 ≡ 11 (mod 13).

If you can share the exact intended expression (including parentheses or operator between 1/2 and 10), I can provide the definitive simplification and align it with the corresponding option label.

MATH3066 (ND) 2025 Mid-Semester Test Practice Set C

Simplify the following expression, working over 𝑍 13 : 3 − 1 2 10 =

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