1. Given the PIN code formula: $$\text{Pin Code} = \frac{5_0'(3x^3 - 2x^2 + x - 4) \times 4^2}{\sqrt{x^2 - 3x + 24}}$$ Here, we interpret $5_0'$ as simply the constant 5 (assuming a typographical notation). 2. Simplify the numerator: Calculate $4^2 = 16$. Numerator becomes: $$5 \times (3x^3 - 2x^2 + x - 4) \times 16 = 80(3x^3 - 2x^2 + x - 4)$$ 3. The expression for the code is now: $$\frac{80(3x^3 - 2x^2 + x - 4)}{\sqrt{x^2 - 3x + 24}}$$ 4. To evaluate the PIN code for a specific value of $x$, substitute $x$ into both numerator and denominator. For example, at $x=2$: - Numerator: $$80(3(2)^3 - 2(2)^2 + 2 - 4) = 80(3 \times 8 - 2 \times 4 + 2 -4) = 80(24 - 8 + 2 - 4) = 80(14) = 1120$$ - Denominator: $$\sqrt{(2)^2 - 3(2) + 24} = \sqrt{4 - 6 + 24} = \sqrt{22}$$ - Therefore: $$\text{Code} = \frac{1120}{\sqrt{22}}$$ 5. Simplify further by rationalizing denominator: $$\frac{1120}{\sqrt{22}} \times \frac{\sqrt{22}}{\sqrt{22}} = \frac{1120 \sqrt{22}}{22} = \frac{560 \sqrt{22}}{11}$$ So, the code at $x=2$ is: $$\frac{560 \sqrt{22}}{11}$$ This can be approximated by a calculator if needed. --- **Final result:** The PIN code formula is: $$\boxed{\frac{80(3x^3 - 2x^2 + x - 4)}{\sqrt{x^2 - 3x + 24}}}$$ which can be evaluated for desired values of $x$.