Algebraic Manipulation

Algebra Level 3

If m − 1 m = 1 m-\dfrac{1}{m}=1 , find m 8 + 1 m 8 m^{8}+\dfrac{1}{m^{8}} .


The answer is 47.

Philip Ramones by Philip Ramones , 19, Philippines

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1 solution

m − 1 m = 1 ( m − 1 m ) 2 = 1 2 m 2 − 2 + 1 m 2 = 1 m 2 + 1 m 2 = 3 ( m 2 + 1 m 2 ) 2 = 3 2 m 4 + 2 + 1 m 4 = 9 m 4 + 1 m 4 = 7 ( m 4 + 1 m 4 ) 2 = 7 2 m 8 + 2 + 1 m 8 = 49 m 8 + 1 m 8 = 47 \begin{aligned} m - \frac 1m & = 1 \\ \left(m - \frac 1m\right)^2 & = 1^2 \\ m^2 -2 + \frac 1{m^2} & = 1 \\ m^2 + \frac 1{m^2} & = 3 \\ \left(m^2 + \frac 1{m^2}\right)^2 & = 3^2 \\ m^4 +2 + \frac 1{m^4} & = 9 \\ m^4 + \frac 1{m^4} & = 7 \\ \left(m^4 + \frac 1{m^4}\right)^2 & = 7^2 \\ m^8 +2 + \frac 1{m^8} & = 49 \\ m^8 + \frac 1{m^8} & = \boxed{47} \end{aligned}

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