Question

a+1/a=2 find a⁴+1/a² and a⁸+1/a⁴​

Answer 1

Answer:

$\sf{1. \ The \ value \ of \ a^{4}+\dfrac{1}{a^{2}} \ is \ 2.}$

$\sf{2. \ The \ value \ of \ a^{8}+\dfrac{1}{a^{4}} \ is \ 2.}$

Given:

$\sf{\leadsto{a+\dfrac{1}{a}=2}}$

To find:

$\sf{The \ value \ of \ a^{4}+\dfrac{1}{a^{2}} \ and \ a^{8}+\dfrac{1}{a^{8}}.}$

Solution:

$\sf{a+\dfrac{1}{a}=2}$

$\sf{\therefore{\dfrac{a^{2}+1}{a}=2}}$

$\sf{\therefore{a^{2}+1=2a}}$

$\sf{\therefore{a^{2}-2a+1=0}}$

$\sf{\therefore{a^{2}-a-a+1=0}}$

$\sf{\therefore{a(a-1)-1(a-1)=0}}$

$\sf{\therefore{(a-1)(a-1)=0}}$

$\sf{\therefore{a=1 \ or \ 1}}$

$\sf{Hence, \ the \ value \ of \ a \ is \ 1.}$

$\sf{\leadsto{a^{4}+\dfrac{1}{a^{2}}}}$

$\sf{But, \ a=1}$

$\sf{\leadsto{1^{4}+\dfrac{1}{1^{2}}}}$

$\sf{\leadsto{1+1}}$

$\sf{\leadsto{2}}$

$\sf{The \ value \ of \ a^{4}+\dfrac{1}{a^{2}} \ is \ 2.}$

_____________________________________

$\sf{\leadsto{a^{8}+\dfrac{1}{a^{4}}}}$

$\sf{But, \ a=1}$

$\sf{\leadsto{1^{8}+\dfrac{1}{1^{4}}}}$

$\sf{\leadsto{1+1}}$

$\sf{\leadsto{2}}$

$\sf\purple{\tt{\therefore{The \ value \ of \ a^{8}+\dfrac{1}{a^{4}} \ is \ 2.}}}$