If \(x-\frac{1}{x}\) =3, find the values of \(x^2+\frac{1}{x^2}\) an

1.	If \(x-\frac{1}{x}\) =3, find the values of \(x^2+\frac{1}{x^2}\) and \(x^4+\frac{1}{x^4}\).
Answer» (i) Given that, x - \(\frac{1}{x}\) = 3 Squaring both sides, we get (x - \(\frac{1}{x}\))²= (3)² x² - 2 × x × \(\frac{1}{x}\) + (\(\frac{1}{x}\))² = 9 x² - 2 + \(\frac{1}{x\times x}\) = 9 x² + \(\frac{1}{x\times x}\) = 11 (ii) Squaring both sides, we get (x² + \(\frac{1}{x\times x}\))² = (11)² (x²)² + 2 × x² × \(\frac{1}{x\times x}\) + (\(\frac{1}{x\times x}\))² = 121 x⁴ + 2 + \(\frac{1}{x\times x\times x\times x}\) = 121 x⁴ + \(\frac{1}{x\times x\times x\times x}\) = 119

If \(x-\frac{1}{x}\) =3, find the values of \(x^2+\frac{1}{x^2}\) and \(x^4+\frac{1}{x^4}\).

Answer»

(i) Given that,

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

Squaring both sides, we get

(x - \(\frac{1}{x}\))²= (3)²

x² - 2 × x × \(\frac{1}{x}\) + (\(\frac{1}{x}\))² = 9

x² - 2 + \(\frac{1}{x\times x}\) = 9

x² + \(\frac{1}{x\times x}\) = 11

(ii) Squaring both sides, we get

(x² + \(\frac{1}{x\times x}\))² = (11)²

(x²)² + 2 × x² × \(\frac{1}{x\times x}\) + (\(\frac{1}{x\times x}\))² = 121

x⁴ + 2 + \(\frac{1}{x\times x\times x\times x}\) = 121

x⁴ + \(\frac{1}{x\times x\times x\times x}\) = 119