1. **Problem Statement:** Given the equation $\left(x + \frac{1}{x}\right)^2 = 5^2$, find the values of expressions involving powers of $x$ and $\frac{1}{x}$. 2. **Step 1: Expand the given equation using the formula $(a+b)^2 = a^2 + 2ab + b^2$.** $$\left(x + \frac{1}{x}\right)^2 = x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 = x^2 + 2 + \frac{1}{x^2}$$ 3. **Step 2: Substitute the right side value and simplify.** $$x^2 + 2 + \frac{1}{x^2} = 25$$ Subtract 2 from both sides: $$x^2 + \frac{1}{x^2} = 25 - 2 = 23$$ 4. **Step 3: To find $x^4 + \frac{1}{x^4}$, square both sides of the equation from Step 2.** $$\left(x^2 + \frac{1}{x^2}\right)^2 = 23^2$$ Using the formula $(a+b)^2 = a^2 + 2ab + b^2$ again: $$x^4 + 2 + \frac{1}{x^4} = 529$$ 5. **Step 4: Solve for $x^4 + \frac{1}{x^4}$.** Subtract 2 from both sides: $$x^4 + \frac{1}{x^4} = 529 - 2 = 527$$ **Final answers:** $$x^2 + \frac{1}{x^2} = 23$$ $$x^4 + \frac{1}{x^4} = 527$$ This method uses algebraic identities and careful substitution to find higher powers of $x$ and $\frac{1}{x}$ from the given expression.