Solving for \(x^3 - 1/x^3\)

We are given the equation \(x - \frac{1}{x} = 8\). We need to find the value of \(x^3 - \frac{1}{x^3}\).

Using the Algebraic Identity

We can use the algebraic identity for the difference of cubes, which is related to the cube of a binomial difference:

The identity is: \(a^3 - b^3 = (a - b)^3 + 3ab(a - b)\)

Let \(a = x\) and \(b = \frac{1}{x}\).

Substitute these into the identity:

\(x^3 - \frac{1}{x^3} = \left(x - \frac{1}{x}\right)^3 + 3(x)\left(\frac{1}{x}\right)\left(x - \frac{1}{x}\right)\)

Substituting Known Values

We know that \(x - \frac{1}{x} = 8\). Also, the term \((x)\left(\frac{1}{x}\right)\) simplifies to 1.

Substitute these values into the equation:

\(x^3 - \frac{1}{x^3} = (8)^3 + 3(1)(8)\)

Calculating the Result

Now, perform the calculations:

• Calculate \(8^3\): \(8 \times 8 \times 8 = 64 \times 8 = 512\).
• Calculate \(3 \times 1 \times 8\): \(3 \times 8 = 24\).

Add the results:

\(x^3 - \frac{1}{x^3} = 512 + 24\)

\(x^3 - \frac{1}{x^3} = 536\)