Finding the Value of x³ + 1/x³ from x² + 1/x²

We are given an equation involving x² and need to find the value of an expression involving x³. This type of problem often requires using algebraic identities to move from a lower power to a higher power.

The given information is:

• \(x^2 + \frac{1}{x^2} = 18\)
• \(x > 0\)

We need to find the value of \(x^3 + \frac{1}{x^3}\).

Step 1: Find the value of \(x + \frac{1}{x}\)

We know the algebraic identity for the square of a sum: \((a+b)^2 = a^2 + b^2 + 2ab\). Let's apply this identity with \(a=x\) and \(b=\frac{1}{x}\).

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

This simplifies to:

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

We are given that \(x^2 + \frac{1}{x^2} = 18\). Substituting this value into the equation:

\(\left(x + \frac{1}{x}\right)^2 = 18 + 2\)

\(\left(x + \frac{1}{x}\right)^2 = 20\)

Now, take the square root of both sides:

\(x + \frac{1}{x} = \pm \sqrt{20}\)

\(x + \frac{1}{x} = \pm \sqrt{4 \times 5}\)

\(x + \frac{1}{x} = \pm 2\sqrt{5}\)

We are given that \(x > 0\). If \(x > 0\), then \(\frac{1}{x}\) is also greater than 0. The sum of two positive numbers must be positive. Therefore, we take the positive value:

\(x + \frac{1}{x} = 2\sqrt{5}\)

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

We know the algebraic identity for the cube of a sum: \((a+b)^3 = a^3 + b^3 + 3ab(a+b)\). Let's apply this identity with \(a=x\) and \(b=\frac{1}{x}\).

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

This simplifies to:

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

We want to find \(x^3 + \frac{1}{x^3}\), so we can rearrange the formula:

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

In Step 1, we found that \(x + \frac{1}{x} = 2\sqrt{5}\). Substitute this value into the rearranged formula:

\(x^3 + \frac{1}{x^3} = \left(2\sqrt{5}\right)^3 - 3\left(2\sqrt{5}\right)\)

Calculate \(\left(2\sqrt{5}\right)^3\):

\(\left(2\sqrt{5}\right)^3 = 2^3 \times \left(\sqrt{5}\right)^3\)

\(\left(2\sqrt{5}\right)^3 = 8 \times \left(\sqrt{5}\right)^2 \times \sqrt{5}\)

\(\left(2\sqrt{5}\right)^3 = 8 \times 5 \times \sqrt{5}\)

\(\left(2\sqrt{5}\right)^3 = 40\sqrt{5}\)

Now substitute this back into the equation for \(x^3 + \frac{1}{x^3}\):

\(x^3 + \frac{1}{x^3} = 40\sqrt{5} - 6\sqrt{5}\)

\(x^3 + \frac{1}{x^3} = (40 - 6)\sqrt{5}\)

\(x^3 + \frac{1}{x^3} = 34\sqrt{5}\)

Summary of Steps

Thus, the value of \(x^3 + \frac{1}{x^3}\) is \(34\sqrt{5}\).

Revision Table: Key Algebraic Identities

Understanding algebraic identities is crucial for solving such problems.

Additional Information on Algebraic Expressions

Algebraic expressions are combinations of variables (like \(x\)) and constants (like 18), connected by mathematical operations (+, -, ×, ÷). Problems involving powers of variables, like finding \(x^3 + \frac{1}{x^3}\) from \(x^2 + \frac{1}{x^2}\), are common in algebra and competitive exams.

When dealing with expressions like \(x^n + \frac{1}{x^n}\) or \(x^n - \frac{1}{x^n}\), algebraic identities are powerful tools. Knowing how to manipulate these identities allows you to find values of higher power expressions if the values of lower power expressions are known, or vice versa.

The condition \(x > 0\) is important because it helps determine the sign when taking a square root. If \(x\) could be negative, \(x + \frac{1}{x}\) could potentially be negative, and we would need to consider both cases for \(x^2 + \frac{1}{x^2}\). However, for \(x > 0\), \(x + \frac{1}{x}\) is always positive.