Solving for x^3 / x^3 Using Given Value and Algebraic Manipulation

When faced with the problem of finding the value of x^3 / x^3 given that x^2 / x^2 83, we can employ a series of algebraic manipulations to simplify and solve the problem. This involves utilizing relationships between the given expression and other algebraic forms, performing substitutions, and solving for the required values.

Step-by-Step Solution

Given: x^2 / x^2 83

Step 1: Deriving the Value of x * 1/x

The first step is to derive the value of x * 1/x from the given equation. We can use the following relationship:

x^2 / x^2 (x * 1/x)^2 - 2

Let's set y x * 1/x. Then the equation becomes:

x^2 / x^2 y^2 - 2

Substitute the known value:

y^2 - 2 83

This simplifies to:

y^2 85

Take the square root of both sides:

y √85 or y -√85

Thus, we have two possible values for x * 1/x based on the given equation.

Step 2: Finding the Value of x^3 / x^3

Next, we need to find the value of x^3 / x^3. We can use the following algebraic identity:

x^3 / x^3 (x * 1/x) * (x^2 / x^2) - (x * 1/x)

Substitute x^2 / x^2 83 and the possible values of x * 1/x:

x^3 / x^3 y * 83 - y 82y

Now, substitute the possible values of y:

If y √85:

x^3 / x^3 82√85

If y -√85:

x^3 / x^3 82(-√85) -82√85

Therefore, the value of x^3 / x^3 is either 82√85 or -82√85, depending on the sign of y.

Final Answer

The final answer for the value of x^3 / x^3 is:

boxed{82√85} or -82√85

This detailed approach highlights the importance of algebraic manipulation and substitution in solving complex mathematical problems.