Solving Algebraic Expressions with Exponents

The problem asks us to find the value of \(x^4 - \frac{1}{x^4}\) given the equation \(x^2 - \frac{1}{x^2} = 4\sqrt{2}\).

We need to evaluate the expression \(x^4 - \frac{1}{x^4}\). Let's first look at the structure of this expression. It is a difference of squares, specifically \((x^2)^2 - (\frac{1}{x^2})^2\).

Using the algebraic identity for the difference of squares, \(a^2 - b^2 = (a-b)(a+b)\), we can factor \(x^4 - \frac{1}{x^4}\) where \(a = x^2\) and \(b = \frac{1}{x^2}\):

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

We are given the value of the first factor, \(x^2 - \frac{1}{x^2} = 4\sqrt{2}\). However, we still need to find the value of the second factor, \(x^2 + \frac{1}{x^2}\).

We can find \(x^2 + \frac{1}{x^2}\) from the given equation \(x^2 - \frac{1}{x^2} = 4\sqrt{2}\) using another algebraic identity that relates the square of a sum and the square of a difference:

\[(a+b)^2 = (a-b)^2 + 4ab\]

Let \(a = x^2\) and \(b = \frac{1}{x^2}\). Then \(ab = x^2 \times \frac{1}{x^2} = 1\). Substituting these into the identity:

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

Substitute the given value \(x^2 - \frac{1}{x^2} = 4\sqrt{2}\) into the equation:

\[\left(x^2 + \frac{1}{x^2}\right)^2 = \left(4\sqrt{2}\right)^2 + 4(1)\] \[\left(x^2 + \frac{1}{x^2}\right)^2 = (4^2 \times (\sqrt{2})^2) + 4\] \[\left(x^2 + \frac{1}{x^2}\right)^2 = (16 \times 2) + 4\] \[\left(x^2 + \frac{1}{x^2}\right)^2 = 32 + 4\] \[\left(x^2 + \frac{1}{x^2}\right)^2 = 36\]

Taking the square root of both sides (assuming \(x^2 + \frac{1}{x^2}\) is positive, which is true for real \(x \neq 0\)):

\[x^2 + \frac{1}{x^2} = \sqrt{36} = 6\]

Now we have the values for both factors in the expression for \(x^4 - \frac{1}{x^4}\):

• \(x^2 - \frac{1}{x^2} = 4\sqrt{2}\) (given)
• \(x^2 + \frac{1}{x^2} = 6\) (calculated)

Substitute these values back into the factored expression:

\[x^4 - \frac{1}{x^4} = \left(x^2 - \frac{1}{x^2}\right)\left(x^2 + \frac{1}{x^2}\right)\] \[x^4 - \frac{1}{x^4} = (4\sqrt{2})(6)\] \[x^4 - \frac{1}{x^4} = 24\sqrt{2}\]

Thus, the value of \(x^4 - \frac{1}{x^4}\) is \(24\sqrt{2}\).

Revision Table: Key Concepts

Let's quickly summarize the main algebraic concepts used:

• Difference of Squares: \(a^2 - b^2 = (a-b)(a+b)\)
• Relationship between Squares of Sum and Difference: \((a+b)^2 = (a-b)^2 + 4ab\)

Additional Information: Algebraic Identities

Algebraic identities are equations that are true for all possible values of the variables involved. They are fundamental tools for simplifying expressions and solving equations. Here are a few common identities:

• \((a+b)^2 = a^2 + 2ab + b^2\)
• \((a-b)^2 = a^2 - 2ab + b^2\)
• \(a^2 - b^2 = (a-b)(a+b)\)
• \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\)
• \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)
• \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
• \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

In this problem, recognizing \(x^4 - \frac{1}{x^4}\) as a difference of squares and knowing how to find \(x^2 + \frac{1}{x^2}\) from \(x^2 - \frac{1}{x^2}\) using the relationship between \((a+b)^2\) and \((a-b)^2\) were crucial steps.