Solving the Mathematical Puzzle: (a^3/b^3) / (b^3/a^3) When a/b - b/a 3

Discover how to solve complex mathematical puzzles, including simplifying algebraic expressions and solving quadratic equations. This guide will help you understand and solve a challenging problem: if (frac{a}{b} - frac{b}{a} 3), what is the value of (frac{a^3}{b^3} div frac{b^3}{a^3})?

Understanding the Problem

The problem requires us to find the value of (frac{a^3}{b^3} div frac{b^3}{a^3}) given the condition (frac{a}{b} - frac{b}{a} 3).

Simplification and Substitution

Let's simplify the expression by letting (x frac{a}{b}). This substitution transforms our given condition into a more manageable equation.

Substitute (x) into the given condition:

[frac{a}{b} - frac{b}{a} 3]

becomes:

[x - frac{1}{x} 3]

Solving the Quadratic Equation

To solve for (x), we will multiply both sides by (x) to eliminate the fraction:

[x^2 - 1 3x]

Bring all terms to one side of the equation to form a standard quadratic equation:

[x^2 - 3x - 1 0]

Solve this quadratic equation using the quadratic formula:

[x frac{-b pm sqrt{b^2 - 4ac}}{2a}]

Here, (a 1), (b -3), and (c -1). Substitute these values into the formula:

[x frac{3 pm sqrt{(-3)^2 - 4 cdot 1 cdot (-1)}}{2 cdot 1} frac{3 pm sqrt{9 4}}{2} frac{3 pm sqrt{13}}{2}]

Evaluating the Expression (frac{a^3}{b^3} div frac{b^3}{a^3})

The next step is to evaluate (frac{a^3}{b^3} div frac{b^3}{a^3}) in terms of (x). This simplifies to:

[x^3 div frac{1}{x^3} x^3 cdot x^3 x^6]

Now, we will use the identity ((x - frac{1}{x})^2 x^2 - 2 frac{1}{x^2}) to find (x^2 frac{1}{x^2}).

Square both sides of the original equation (x - frac{1}{x} 3):

[(x - frac{1}{x})^2 3^2]

This yields:

[x^2 - 2 frac{1}{x^2} 9]

Rearrange to find (x^2 frac{1}{x^2}):

[x^2 frac{1}{x^2} 11]

The value of (x frac{1}{x}) can be found from the identity ((x frac{1}{x})^2 x^2 frac{1}{x^2} 2):

[(x frac{1}{x})^2 11 2 13]

Therefore:

[x frac{1}{x} sqrt{13}]

Substitute this back into the expression for (x^3 - frac{1}{x^3}):

[x^3 - frac{1}{x^3} left(x frac{1}{x}right)^3 - 3left(x frac{1}{x}right) 13sqrt{13} - 3sqrt{13} 10sqrt{13}]

Hence, the value of (frac{a^3}{b^3} div frac{b^3}{a^3} 10sqrt{13}).

Final Answer

The final value is:

[boxed{10sqrt{13}}]

Concluding Thoughts

This problem showcases the power of algebraic manipulation and the application of quadratic equations. By breaking down the problem into simpler steps, we can solve complex mathematical puzzles efficiently. This process not only helps in solving the problem but also improves your problem-solving skills in algebra.