Calculating Algebraic Expression Values: \(x^8 - \frac{1}{x^8}\)

This problem requires us to find the value of a higher-power algebraic expression given the value of a lower-power expression involving the same variable. We are given the value of \((x - \frac{1}{x})\) and need to find the value of \((x^8 - \frac{1}{x^8})\). We will use fundamental algebraic identities to solve this step by step.

Understanding the Given Information

We are given the following:

• The equation: \(x - \frac{1}{x} = \sqrt{6}\)
• The condition: \(x > 1\) (This helps us determine the sign of expressions like \(x + \frac{1}{x}\)).

Our Goal

We need to calculate the value of the expression \((x^8 - \frac{1}{x^8})\).

Key Algebraic Identities

We will use the following identities repeatedly:

• Difference of Squares: \(a^2 - b^2 = (a - b)(a + b)\)
• Squaring a Difference: \((a - b)^2 = a^2 - 2ab + b^2\)
• Squaring a Sum: \((a + b)^2 = a^2 + 2ab + b^2\)

Specifically for terms involving \(x\) and \(\frac{1}{x}\):

• \((x - \frac{1}{x})^2 = x^2 - 2(x)(\frac{1}{x}) + (\frac{1}{x})^2 = x^2 - 2 + \frac{1}{x^2}\)
• \((x + \frac{1}{x})^2 = x^2 + 2(x)(\frac{1}{x}) + (\frac{1}{x})^2 = x^2 + 2 + \frac{1}{x^2}\)
• From the above, we can see that \((x + \frac{1}{x})^2 - (x - \frac{1}{x})^2 = (x^2 + 2 + \frac{1}{x^2}) - (x^2 - 2 + \frac{1}{x^2}) = 4\). So, \((x + \frac{1}{x})^2 = (x - \frac{1}{x})^2 + 4\).
• \((x^n - \frac{1}{x^n}) = (x^{n/2} - \frac{1}{x^{n/2}})(x^{n/2} + \frac{1}{x^{n/2}})\)

Step-by-Step Calculation of \(x^8 - \frac{1}{x^8}\)

Step 1: Finding \(x + \frac{1}{x}\)

We are given \(x - \frac{1}{x} = \sqrt{6}\). We can find \(x + \frac{1}{x}\) using the identity \((x + \frac{1}{x})^2 = (x - \frac{1}{x})^2 + 4\).

Substitute the given value:

\((x + \frac{1}{x})^2 = (\sqrt{6})^2 + 4\)

\((x + \frac{1}{x})^2 = 6 + 4\)

\((x + \frac{1}{x})^2 = 10\)

Taking the square root of both sides, we get \(x + \frac{1}{x} = \pm\sqrt{10}\). Since we are given \(x > 1\), both \(x\) and \(\frac{1}{x}\) are positive, so their sum must be positive. Therefore:

\(x + \frac{1}{x} = \sqrt{10}\)

Step 2: Finding \(x^2 - \frac{1}{x^2}\)

We can find \(x^2 - \frac{1}{x^2}\) using the difference of squares identity: \(a^2 - b^2 = (a - b)(a + b)\). Here \(a = x\) and \(b = \frac{1}{x}\).

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

Substitute the values we found for \((x - \frac{1}{x})\) and \((x + \frac{1}{x})\):

\(x^2 - \frac{1}{x^2} = (\sqrt{6})(\sqrt{10})\)

\(x^2 - \frac{1}{x^2} = \sqrt{6 \times 10} = \sqrt{60}\)

Simplify the square root:

\(\sqrt{60} = \sqrt{4 \times 15} = \sqrt{4} \times \sqrt{15} = 2\sqrt{15}\)

So,

\(x^2 - \frac{1}{x^2} = 2\sqrt{15}\)

Step 3: Finding \(x^2 + \frac{1}{x^2}\)

We can find \(x^2 + \frac{1}{x^2}\) by squaring either \((x - \frac{1}{x})\) or \((x + \frac{1}{x})\).

Using \((x - \frac{1}{x}) = \sqrt{6}\):

\((x - \frac{1}{x})^2 = (\sqrt{6})^2\)

\(x^2 - 2 + \frac{1}{x^2} = 6\)

\(x^2 + \frac{1}{x^2} = 6 + 2\)

\(x^2 + \frac{1}{x^2} = 8\)

Let's verify using \((x + \frac{1}{x}) = \sqrt{10}\):

\((x + \frac{1}{x})^2 = (\sqrt{10})^2\)

\(x^2 + 2 + \frac{1}{x^2} = 10\)

\(x^2 + \frac{1}{x^2} = 10 - 2\)

\(x^2 + \frac{1}{x^2} = 8\)

Both methods give the same result.

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

Again, use the difference of squares identity: \(a^2 - b^2 = (a - b)(a + b)\). Here \(a = x^2\) and \(b = \frac{1}{x^2}\).

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

Substitute the values we found in Step 2 and Step 3:

\(x^4 - \frac{1}{x^4} = (2\sqrt{15})(8)\)

\(x^4 - \frac{1}{x^4} = 16\sqrt{15}\)

Step 5: Finding \(x^4 + \frac{1}{x^4}\)

We can find \(x^4 + \frac{1}{x^4}\) by squaring \(x^2 + \frac{1}{x^2}\).

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

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

Substitute the value of \(x^2 + \frac{1}{x^2}\) from Step 3:

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

\(64 = x^4 + 2 + \frac{1}{x^4}\)

\(x^4 + \frac{1}{x^4} = 64 - 2\)

\(x^4 + \frac{1}{x^4} = 62\)

Step 6: Finding \(x^8 - \frac{1}{x^8}\)

Finally, use the difference of squares identity again: \(a^2 - b^2 = (a - b)(a + b)\). Here \(a = x^4\) and \(b = \frac{1}{x^4}\).

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

Substitute the values we found in Step 4 and Step 5:

\(x^8 - \frac{1}{x^8} = (16\sqrt{15})(62)\)

Now, perform the multiplication:

\(16 \times 62 = 16 \times (60 + 2) = 16 \times 60 + 16 \times 2 = 960 + 32 = 992\)

So,

\(x^8 - \frac{1}{x^8} = 992\sqrt{15}\)

Summary of Intermediate Results

Conclusion on Algebraic Expression Value

Starting from the given value of \((x - \frac{1}{x})\), and using algebraic identities, we have successfully calculated the value of \((x^8 - \frac{1}{x^8})\) to be \(992\sqrt{15}\).

Revision Table: Algebraic Identities Practice

Regular practice with algebraic identities is key to solving problems like this quickly and accurately. Here's a quick reference:

Additional Information: Powers of x and 1/x

When dealing with expressions like \(x^n \pm \frac{1}{x^n}\), notice the pattern that emerges when you square them:

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

This pattern allows us to move from a lower power (\(n\)) to a higher power (\(2n\))) for the sum form (\(x^{2n} + \frac{1}{x^{2n}}\)).

To get the difference form (\(x^{2n} - \frac{1}{x^{2n}}\)), we use the difference of squares identity, which requires both the sum and difference of the lower powers:

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

In this problem, we started with \(x - \frac{1}{x}\) (power 1), calculated \(x + \frac{1}{x}\) (power 1), then used these to find \(x^2 - \frac{1}{x^2}\) (power 2). We also calculated \(x^2 + \frac{1}{x^2}\) (power 2) by squaring the power 1 terms. We then used the power 2 terms (\(x^2 - \frac{1}{x^2}\) and \(x^2 + \frac{1}{x^2}\)) to find the power 4 terms (\(x^4 - \frac{1}{x^4}\) and \(x^4 + \frac{1}{x^4}\)). Finally, we used the power 4 terms to find the power 8 term (\(x^8 - \frac{1}{x^8}\)). This step-by-step approach is crucial for solving such problems.