We are given the equation:

\[
x + \frac{1}{x} = 52
\]

We are tasked with finding:

\[
x^2 + \frac{1}{x^2} + \left(\frac{1}{x}\right)^3
\]

### Step 1: Square both sides of the original equation
Start by squaring both sides of \(x + \frac{1}{x} = 52\):

\[
\left( x + \frac{1}{x} \right)^2 = 52^2
\]

Expanding the left-hand side using the identity \((a+b)^2 = a^2 + 2ab + b^2\):

\[
x^2 + 2 + \frac{1}{x^2} = 2704
\]

Subtract 2 from both sides:

\[
x^2 + \frac{1}{x^2} = 2702
\]

### Step 2: Find \(\left(\frac{1}{x}\right)^3\)
From the original equation \(x + \frac{1}{x} = 52\), cube both sides:

\[
\left(x + \frac{1}{x}\right)^3 = 52^3
\]

Expand the left-hand side using the identity \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\), which simplifies to:

\[
x^3 + 3x + 3\frac{1}{x} + \frac{1}{x^3} = 52^3
\]

But we can simplify this further without solving the full cubic:

### Step 3: Add the expressions
Thus, the final expression to solve:
\[
x^2 + \frac

x + 1/x = 52 then x² + 1/x² + (1/x)³ = ?

Solve the equation x + 1/x = 52 and find x² + 1/x² + (1/x)³. This step-by-step guide covers algebraic concepts including squaring and cubing binomials, powers, and exponents. Ideal for high school students in Grades 10-12.

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