Question

if x + 1/x = 2. find the value of x⁸⁹ + 1/x²⁰⁹⁷ - x⁶⁴⁹​

Answer 1

given

X+ 1/X = 2

x^2- 2x + 1 = 0

(X - 1)^2= 0

so

X = 1

Now

x⁸⁹ + 1/x²⁰⁹⁷ - x⁶⁴⁹

on putting X = 1 we get

1⁸⁹ + 1/1²⁰⁹⁷ - 1⁶⁴⁹

1 + 1 -1

2-1

1

Answer 2

$\huge\star{\green{\underline{\mathfrak{Answer :-}}}}$

Given:

$\hookrightarrow\sf{x+\dfrac{1}{x}=2}$

To Find:

$\hookrightarrow\sf{{x}^{89}+\dfrac{1}{{x}^{2097}}-{{x}^{649}}}$

Solution:

$\hookrightarrow\sf{x+\dfrac{1}{x}=2}$

Solving this,

$\hookrightarrow\sf{\dfrac{{x}^{2}+1}{x}=2}$

$\hookrightarrow\sf{{x}^{2}+1=2x}$

$\hookrightarrow\sf{{x}^{2}-2x+1=0}$

$\hookrightarrow\sf{{(x-1)}^{2}=0}$

This is because, (a+b)² = a² + b² + 2ab.

Solving further,

$\hookrightarrow\underline{\sf{x=1}}$

Now, using this.

$\hookrightarrow\sf{{x}^{89}+\dfrac{1}{{x}^{2097}}-{{x}^{649}}}$

$\hookrightarrow\sf{{1}^{89}+\dfrac{1}{{1}^{2097}}-{{1}^{649}}}$

[ This is because 1 to the power anything is always 1 ]

$\hookrightarrow\sf{1+\dfrac{1}{1}-1}$

$\hookrightarrow\sf{\dfrac{1+1-1}{1}}$

$\hookrightarrow\sf{\dfrac{1}{1}}$

$\huge\hookrightarrow\underline{\sf{Answer:\:1}}$

$\rule{200}4$