Question

If $m-\frac{1}{m}=5,$ find :

(i) $m^2+\frac{1}{m^2}$

(ii) $m^4+\frac{1}{m^4}$

(iii) $m^2-\frac{1}{m^2}$

Answer 1

(i) $m^2+\frac{1}{m^2}={(m-\frac{1}{m})}^2+2=(5{)}^2+2=25+2=27$

(ii) $m^4+\frac{1}{m^4}={(m^2+\frac{1}{m^2})}^2-2=(27{)}^2-2=729-2=727$

(iii) $m^2-\frac{1}{m^2}=(m+\frac{1}{m})(m-\frac{1}{m})=5(m+\frac{1}{m})Now{(m+\frac{1}{m})}^2={(m-\frac{1}{m})}^2+4=(5{)}^2+4=25+4=29\therefore m+\frac{1}{m}=\sqrt{29}\therefore m^2-\frac{1}{m^2}=(5\left)\right(\sqrt{29})5\sqrt{29}$