If (x^2 + 1/x^2) = 51, find the value of x^3 – 1/x^3.

Question

If \((x^2 + \frac{1}{x^2}) = 51\), find the value of \(x^3 - \frac{1}{x^3}\).

Answer 1

We know that: (x – y)² = x² + y² – 2xy 

Replace y with \(\frac{1}{x}\), we get 

\((x - \frac{1}{x})^2\) = x² + \(\frac{1}{x^2}\) – 2 

Since \((x^2 + \frac{1}{x^2}) = 51\)

\((x - \frac{1}{x})^2\) = 51-2 = 49

\((x - \frac{1}{x})\) = ±7 

Now, Find  \(x^3 - \frac{1}{x^3}\)

We know that, x³ – y³ = (x – y)(x² + y² + xy)

Replace y with \(\frac{1}{x}\) , we get 

x³ – \(\frac{1}{x^3}\) = (x – \(\frac{1}{x}\))(x² + \(\frac{1}{x^2}\) + 1) 

Use (x – \(\frac{1}{x}\)) = 7 and (x² + \(\frac{1}{x^2}\)) = 51 

 x³ – \(\frac{1}{x^3}\) = 7 x 52

x³ – \(\frac{1}{x^3}\) = 364