Determine without the use of a calculator: \( \frac{2023}{2022^{3}-1} \)

To simplify the expression \( \frac{2023}{2022^{3}-1} \), we can start by factoring the denominator. We notice that \( 2022^{3} - 1 \) is a difference of cubes, which can be factored using the identity \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \). Here, we can set \( a = 2022 \) and \( b = 1 \): \[ 2022^{3} - 1^{3} = (2022 - 1)(2022^{2} + 2022 \cdot 1 + 1^{2}) = 2021(2022^{2} + 2022 + 1) \] Next, we need to calculate \( 2022^{2} + 2022 + 1 \): \[ 2022^{2} = 4088484 \] So, \[ 2022^{2} + 2022 + 1 = 4088484 + 2022 + 1 = 4088484 + 2023 = 4088507 \] The denominator now becomes: \[ 2022^{3} - 1 = 2021 \times 4088507 \] Thus, we can rewrite the original expression: \[ \frac{2023}{2022^{3)-1}} = \frac{2023}{2021 \times 4088507} \] At this stage, we notice that \( 2023 = 2021 + 2 \). Hence: \[ \frac{2023}{2021 \times 4088507} = \frac{2021 + 2}{2021 \times 4088507} \] This can be separated into two fractions: \[ \frac{2021}{2021 \times 4088507} + \frac{2}{2021 \times 4088507} = \frac{1}{4088507} + \frac{2}{2021 \times 4088507} \] Therefore, the simplified form of the fraction can be expressed as: \[ \frac{1}{4088507} + \frac{2}{2021 \times 4088507} \] Thus, while the final simplified version is cumbersome, the essence of the problem reduces it to straightforward calculations based on factoring and simple arithmetic operations, reinforcing the power of algebraic identities in simplifying rational expressions. The final answer to the problem is more straightforwardly evaluated if required: \[ = \frac{2023}{2021(2022^2 + 2022 + 1)} \] This represents a clean representation of the expression without further need for calculation unless a numerical approximation was requested.