Correct Answer: B) 123 To determine the value of $x^5 + \frac{1}{x^5}$, we will leverage the given equation $x + \frac{1}{x} = 3$ and fundamental algebraic identities. **Step 1: Calculate $x^2 + \frac{1}{x^2}$** We know that $(x + \frac{1}{x})^2 = x^2 + 2(x)(\frac{1}{x}) + \frac{1}{x^2} = x^2 + \frac{1}{x^2} + 2$. Substituting the given value: $3^2 = x^2 + \frac{1}{x^2} + 2$ $9 = x^2 + \frac{1}{x^2} + 2$ $x^2 + \frac{1}{x^2} = 9 - 2 = 7$. (Equation 1) **Step 2: Calculate $x^3 + \frac{1}{x^3}$** We know that $(x + \frac{1}{x})^3 = x^3 + 3(x)(\frac{1}{x})(x + \frac{1}{x}) + \frac{1}{x^3} = x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x})$. Substituting the given value: $3^3 = x^3 + \frac{1}{x^3} + 3(3)$ $27 = x^3 + \frac{1}{x^3} + 9$ $x^3 + \frac{1}{x^3} = 27 - 9 = 18$. (Equation 2) **Step 3: Combine the results from Step 1 and Step 2** Multiply Equation 1 by Equation 2: $(x^2 + \frac{1}{x^2})(x^3 + \frac{1}{x^3}) = (7)(18)$ Expanding the left side: $x^2 \cdot x^3 + x^2 \cdot \frac{1}{x^3} + \frac{1}{x^2} \cdot x^3 + \frac{1}{x^2} \cdot \frac{1}{x^3} = x^5 + \frac{1}{x} + x + \frac{1}{x^5}$ Rearranging the terms, we get: $(x^5 + \frac{1}{x^5}) + (x + \frac{1}{x})$ So, we have: $(x^5 + \frac{1}{x^5}) + (x + \frac{1}{x}) = 126$ **Step 4: Solve for $x^5 + \frac{1}{x^5}$** Substitute the original given value $x + \frac{1}{x} = 3$ into the equation from Step 3: $(x^5 + \frac{1}{x^5}) + 3 = 126$ $x^5 + \frac{1}{x^5} = 126 - 3$ $x^5 + \frac{1}{x^5} = 123$ Thus, the value of $x^5 + \frac{1}{x^5}$ is 123.