Simplify the following product: \(\left( {x + \frac{1}{x}} \right)\left( {x - \frac{1}{x}} \right)\left( {{x^2} + \frac{1}{{{x^2}}}} \right)\)
The correct answer is {{x^4} - \frac{1}{{{x^4}}}}.
In this specific problem, we could have expanded directly, but recognizing the difference of squares pattern significantly simplified the process. Direct expansion of \left( {x + \frac{1}{x}} \right)\left( {x - \frac{1}{x}} \right) gives x \cdot x - x \cdot \frac{1}{x} + \frac{1}{x} \cdot x - \frac{1}{x} \cdot \frac{1}{x} = {x^2} - 1 + 1 - \frac{1}{{{x^2}}} = {x^2} - \frac{1}{{{x^2}}}, confirming the identity approach. Always look for opportunities to use identities as they simplify complex multiplications.
Similar Questions
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Important Questions from Identities
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