Question:
If \[ \frac ab+\frac ba=2 \] find:
\[ \left(\frac ab\right)^{100} – \left(\frac ba\right)^{100} \]
Solution:
Using identity:
\[ \left(x-y\right)^2 = \left(x+y\right)^2-4xy \]
Let \[ x=\frac ab,\qquad y=\frac ba \]
Then,
\[ xy=\frac ab\cdot\frac ba=1 \]
Given:
\[ x+y=2 \]
Therefore,
\[ (x-y)^2=(2)^2-4(1) \]
\[ =4-4 \]
\[ =0 \]
\[ x-y=0 \]
\[ x=y \]
\[ \frac ab=\frac ba \]
Hence,
\[ \left(\frac ab\right)^{100} = \left(\frac ba\right)^{100} \]
Therefore,
\[ \left(\frac ab\right)^{100} – \left(\frac ba\right)^{100} = 0 \]