Solving for \(x^8 - \frac{1}{x^8}\) given \(x + \frac{1}{x} = 5\)

The problem asks us to find the value of the expression \(x^8 - \frac{1}{x^8}\) given that \(x + \frac{1}{x} = 5\) and \(x > 1\).

We can solve this problem by using algebraic identities to find intermediate terms like \(x^2 + \frac{1}{x^2}\), \(x^4 + \frac{1}{x^4}\), and \(x - \frac{1}{x}\). The expression \(x^8 - \frac{1}{x^8}\) can be factored using the difference of squares formula, \(a^2 - b^2 = (a+b)(a-b)\).

Step-by-Step Calculation

Step 1: Find \(x^2 + \frac{1}{x^2}\)

We are given \(x + \frac{1}{x} = 5\). Squaring both sides, we get:

\(\left(x + \frac{1}{x}\right)^2 = 5^2\)

Using the identity \((a+b)^2 = a^2 + 2ab + b^2\):

\(x^2 + 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2 = 25\)

\(x^2 + 2 + \frac{1}{x^2} = 25\)

Subtracting 2 from both sides:

\(x^2 + \frac{1}{x^2} = 25 - 2\)

\(x^2 + \frac{1}{x^2} = 23\)

Step 2: Find \(x^4 + \frac{1}{x^4}\)

Now we use the value of \(x^2 + \frac{1}{x^2}\). Squaring both sides of \(x^2 + \frac{1}{x^2} = 23\):

\(\left(x^2 + \frac{1}{x^2}\right)^2 = 23^2\)

Using the identity \((a+b)^2 = a^2 + 2ab + b^2\) again:

\((x^2)^2 + 2 \cdot x^2 \cdot \frac{1}{x^2} + \left(\frac{1}{x^2}\right)^2 = 529\)

\(x^4 + 2 + \frac{1}{x^4} = 529\)

Subtracting 2 from both sides:

\(x^4 + \frac{1}{x^4} = 529 - 2\)

\(x^4 + \frac{1}{x^4} = 527\)

Step 3: Find \(x - \frac{1}{x}\)

We can find \(x - \frac{1}{x}\) using the identity \((a-b)^2 = a^2 - 2ab + b^2\).

\(\left(x - \frac{1}{x}\right)^2 = x^2 - 2 \cdot x \cdot \frac{1}{x} + \left(\frac{1}{x}\right)^2\)

\(\left(x - \frac{1}{x}\right)^2 = x^2 - 2 + \frac{1}{x^2}\)

We already know \(x^2 + \frac{1}{x^2} = 23\). Substitute this value:

\(\left(x - \frac{1}{x}\right)^2 = \left(x^2 + \frac{1}{x^2}\right) - 2\)

\(\left(x - \frac{1}{x}\right)^2 = 23 - 2\)

\(\left(x - \frac{1}{x}\right)^2 = 21\)

Taking the square root of both sides:

\(x - \frac{1}{x} = \pm \sqrt{21}\)

The problem states that \(x > 1\). If \(x > 1\), then \(\frac{1}{x} < 1\), and thus \(x - \frac{1}{x}\) must be positive. So, we take the positive square root:

\(x - \frac{1}{x} = \sqrt{21}\)

Step 4: Calculate \(x^8 - \frac{1}{x^8}\)

We factor the expression \(x^8 - \frac{1}{x^8}\) repeatedly using the difference of squares identity \(a^2 - b^2 = (a+b)(a-b)\).

\(x^8 - \frac{1}{x^8} = (x^4)^2 - \left(\frac{1}{x^4}\right)^2 = \left(x^4 + \frac{1}{x^4}\right)\left(x^4 - \frac{1}{x^4}\right)\)

\(x^4 - \frac{1}{x^4} = (x^2)^2 - \left(\frac{1}{x^2}\right)^2 = \left(x^2 + \frac{1}{x^2}\right)\left(x^2 - \frac{1}{x^2}\right)\)

\(x^2 - \frac{1}{x^2} = x^2 - \left(\frac{1}{x}\right)^2 = \left(x + \frac{1}{x}\right)\left(x - \frac{1}{x}\right)\)

Combining these factorizations, we get:

\(x^8 - \frac{1}{x^8} = \left(x^4 + \frac{1}{x^4}\right)\left(x^2 + \frac{1}{x^2}\right)\left(x + \frac{1}{x}\right)\left(x - \frac{1}{x}\right)\)

Now, substitute the values we found in the previous steps:

• \(x + \frac{1}{x} = 5\) (Given)
• \(x - \frac{1}{x} = \sqrt{21}\) (From Step 3)
• \(x^2 + \frac{1}{x^2} = 23\) (From Step 1)
• \(x^4 + \frac{1}{x^4} = 527\) (From Step 2)

Substitute these into the factored expression:

\(x^8 - \frac{1}{x^8} = (527)(23)(5)(\sqrt{21})\)

Now, perform the multiplication:

\(527 \times 23 = 12121\)

\(12121 \times 5 = 60605\)

So, \(x^8 - \frac{1}{x^8} = 60605\sqrt{21}\).

Summary of Values

The calculated value \(60605\sqrt{21}\) matches one of the given options.

Revision Table: Key Algebraic Identities

Additional Information: Understanding the \(x > 1\) Condition

The condition \(x > 1\) is important when we calculate \(x - \frac{1}{x}\).

• If \(x > 1\), then \(x\) is a positive number greater than 1.
• The reciprocal \(\frac{1}{x}\) will be a positive number between 0 and 1.
• For example, if \(x=2\), \(\frac{1}{x}=0.5\), and \(x - \frac{1}{x} = 2 - 0.5 = 1.5\) (positive).
• If \(x=0.5\) (which violates \(x>1\)), \(\frac{1}{x}=2\), and \(x - \frac{1}{x} = 0.5 - 2 = -1.5\) (negative).

When we found \(\left(x - \frac{1}{x}\right)^2 = 21\), taking the square root gave us two possibilities: \(\sqrt{21}\) and \(-\sqrt{21}\). The condition \(x > 1\) tells us that \(x - \frac{1}{x}\) must be positive, so we correctly chose \(\sqrt{21}\).

This example shows how understanding the constraints given in a problem, like \(x > 1\), is crucial for selecting the correct value when square roots are involved.