Certainly! Here is a detailed, step-by-step solution to find \(x^2 + \frac{1}{x^2}\) and \(x^4 + \frac{1}{x^4}\) given that \(x - \frac{1}{x} = 5\).
### Step 1: Square the given equation
Given:
[tex]\[ x - \frac{1}{x} = 5 \][/tex]
Square both sides of the equation:
[tex]\[ \left( x - \frac{1}{x} \right)^2 = 5^2 \][/tex]
Expand the left-hand side:
[tex]\[ x^2 - 2 \cdot x \cdot \frac{1}{x} + \left( \frac{1}{x} \right)^2 = 25 \][/tex]
Simplify:
[tex]\[ x^2 - 2 + \frac{1}{x^2} = 25 \][/tex]
### Step 2: Isolate \(x^2 + \frac{1}{x^2}\)
Add 2 to both sides to isolate the term we need:
[tex]\[ x^2 + \frac{1}{x^2} = 25 + 2 \][/tex]
[tex]\[ x^2 + \frac{1}{x^2} = 27 \][/tex]
### Step 3: Square the result to find \(x^4 + \frac{1}{x^4}\)
Next, we need to find \( x^4 + \frac{1}{x^4} \).
Square the equation we found in Step 2:
[tex]\[ \left( x^2 + \frac{1}{x^2} \right)^2 = 27^2 \][/tex]
Expand the left-hand side:
[tex]\[ x^4 + 2 \cdot x^2 \cdot \frac{1}{x^2} + \frac{1}{x^4} = 729 \][/tex]
Simplify:
[tex]\[ x^4 + 2 + \frac{1}{x^4} = 729 \][/tex]
### Step 4: Isolate \(x^4 + \frac{1}{x^4}\)
Subtract 2 from both sides to isolate the term we need:
[tex]\[ x^4 + \frac{1}{x^4} = 729 - 2 \][/tex]
[tex]\[ x^4 + \frac{1}{x^4} = 727 \][/tex]
### Summary
- The value of \( x^2 + \frac{1}{x^2} \) is \(27\).
- The value of \( x^4 + \frac{1}{x^4} \) is \(727\).
So, we have:
[tex]\[ x^2 + \frac{1}{x^2} = 27 \][/tex]
[tex]\[ x^4 + \frac{1}{x^4} = 727 \][/tex]
