x-a-2-1-1-x-b-2-1-2-1-x-a-b-1-x-b-a-1-2-x-a-x-b-2-1-2-1-x-a-b-2-1-2-1-1-x-a-b-2-1-2-1-

Question Number 167492 by mathlove last updated on 18/Mar/22

x^{a} = \sqrt{2} + 1 \dots \dots \dots (1)

x^{b} = \sqrt{2} - 1 \dots \dots \dots (2)

\frac{1}{x^{a - b}} + \frac{1}{x^{b - a}} = ?

(1) \div (2) \Rightarrow \frac{x^{a}}{x^{b}} = \frac{\sqrt{2} + 1}{\sqrt{2 - 1}} \Rightarrow x^{a - b} = \frac{\sqrt{2} + 1}{\sqrt{2 - 1}} \Rightarrow \frac{1}{x^{a - b}} = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \dots . (3)

(2) \div (1) \Rightarrow \frac{x^{b}}{x^{a}} = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \Rightarrow x^{b - a} = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} \Rightarrow \frac{1}{x^{b - a}} = \frac{\sqrt{2} + 1}{\sqrt{2} - 1} \dots . (4)

\frac{1}{x^{a - b}} + \frac{1}{x^{b - a}} = \frac{\sqrt{2} - 1}{\sqrt{2} + 1} + \frac{\sqrt{2} + 1}{\sqrt{2} - 1} = \frac{(\sqrt{2} - 1) (\sqrt{2} - 1) + (\sqrt{2} + 1) (\sqrt{2} + 1)}{(\sqrt{2} + 1) (\sqrt{2} - 1)}

= \frac{{(\sqrt{2} - 1)}^{2} + {(\sqrt{2} + 1)}^{2}}{2 - 1} = 2 - 2 \sqrt{2} + 1 + 2 + 2 \sqrt{2} + 1

\frac{1}{x^{a - b}} + \frac{1}{x^{b - a}} = 6

Answered by Rasheed.Sindhi last updated on 18/Mar/22

x^{a} = \sqrt{2} + 1 \dots \dots \dots (1)

x^{b} = \sqrt{2} - 1 \dots \dots \dots (2)

\frac{1}{x^{a - b}} + \frac{1}{x^{b - a}} = ?

\frac{1}{x^{a - b}} + \frac{1}{x^{b - a}} = \frac{1}{\frac{x^{a}}{x^{b}}} + \frac{1}{\frac{x^{b}}{x^{a}}} = \frac{x^{b}}{x^{a}} + \frac{x^{a}}{x^{b}}

= \frac{{(x^{b})}^{2} + {(x^{a})}^{2}}{x^{a} x^{b}}

= \frac{{(\sqrt{2} - 1)}^{2} + {(\sqrt{2} + 1)}^{2}}{(\sqrt{2} + 1) (\sqrt{2} - 1)}

= \frac{2 {(\sqrt{2})}^{2} + 2 {(1)}^{2}}{{(\sqrt{2})}^{2} - {(1)}^{2}}

= \frac{4 + 2}{2 - 1}

= 6