Use Product Rule: \({x}^{a}{x}^{b}={x}^{a+b}\).
\[{s}^{2}{e}^{2}r\imath \times 8x=x-{x}^{2}\imath n(-1,1)\]
Regroup terms.
\[8{e}^{2}\imath {s}^{2}rx=x-{x}^{2}\imath n(-1,1)\]
Simplify \({x}^{2}\imath n(-1,1)\) to \({x}^{2}\imath n\times -1,1\).
\[8{e}^{2}\imath {s}^{2}rx=x-({x}^{2}\imath n\times -1,1)\]
Simplify \({x}^{2}\imath n\times -1\) to \(-{x}^{2}\imath n\).
\[8{e}^{2}\imath {s}^{2}rx=x-(-{x}^{2}\imath n,1)\]
Regroup terms.
\[8{e}^{2}\imath {s}^{2}rx=x-(-\imath {x}^{2}n,1)\]
Simplify \(x-(-\imath {x}^{2}n,1)\) to \(x--\imath {x}^{2}n,1\).
\[8{e}^{2}\imath {s}^{2}rx=x+\imath {x}^{2}n,1\]
Break down the problem into these 2 equations.
\[8{e}^{2}\imath {s}^{2}rx=x+\imath {x}^{2}n\]
\[8{e}^{2}\imath {s}^{2}rx=1\]
Solve the 1st equation: \(8{e}^{2}\imath {s}^{2}rx=x+\imath {x}^{2}n\).
Divide both sides by \(8\).
\[{e}^{2}\imath {s}^{2}rx=\frac{x+\imath {x}^{2}n}{8}\]
Factor out the common term \(x\).
\[{e}^{2}\imath {s}^{2}rx=\frac{x(1+\imath xn)}{8}\]
Divide both sides by \({e}^{2}\).
\[\imath {s}^{2}rx=\frac{\frac{x(1+\imath xn)}{8}}{{e}^{2}}\]
Simplify \(\frac{\frac{x(1+\imath xn)}{8}}{{e}^{2}}\) to \(\frac{x(1+\imath xn)}{8{e}^{2}}\).
\[\imath {s}^{2}rx=\frac{x(1+\imath xn)}{8{e}^{2}}\]
Divide both sides by \(\imath \).
\[{s}^{2}rx=\frac{\frac{x(1+\imath xn)}{8{e}^{2}}}{\imath }\]
Simplify \(\frac{\frac{x(1+\imath xn)}{8{e}^{2}}}{\imath }\) to \(\frac{x(1+\imath xn)}{8{e}^{2}\imath }\).
\[{s}^{2}rx=\frac{x(1+\imath xn)}{8{e}^{2}\imath }\]
Divide both sides by \({s}^{2}\).
\[rx=\frac{\frac{x(1+\imath xn)}{8{e}^{2}\imath }}{{s}^{2}}\]
Simplify \(\frac{\frac{x(1+\imath xn)}{8{e}^{2}\imath }}{{s}^{2}}\) to \(\frac{x(1+\imath xn)}{8{e}^{2}\imath {s}^{2}}\).
\[rx=\frac{x(1+\imath xn)}{8{e}^{2}\imath {s}^{2}}\]
Divide both sides by \(x\).
\[r=\frac{\frac{x(1+\imath xn)}{8{e}^{2}\imath {s}^{2}}}{x}\]
Simplify \(\frac{\frac{x(1+\imath xn)}{8{e}^{2}\imath {s}^{2}}}{x}\) to \(\frac{x(1+\imath xn)}{8{e}^{2}\imath {s}^{2}x}\).
\[r=\frac{x(1+\imath xn)}{8{e}^{2}\imath {s}^{2}x}\]
Cancel \(x\).
\[r=\frac{1+\imath xn}{8{e}^{2}\imath {s}^{2}}\]
\[r=\frac{1+\imath xn}{8{e}^{2}\imath {s}^{2}}\]
Solve the 2nd equation: \(8{e}^{2}\imath {s}^{2}rx=1\).
Divide both sides by \(8\).
\[{e}^{2}\imath {s}^{2}rx=\frac{1}{8}\]
Divide both sides by \({e}^{2}\).
\[\imath {s}^{2}rx=\frac{\frac{1}{8}}{{e}^{2}}\]
Simplify \(\frac{\frac{1}{8}}{{e}^{2}}\) to \(\frac{1}{8{e}^{2}}\).
\[\imath {s}^{2}rx=\frac{1}{8{e}^{2}}\]
Divide both sides by \(\imath \).
\[{s}^{2}rx=\frac{\frac{1}{8{e}^{2}}}{\imath }\]
Simplify \(\frac{\frac{1}{8{e}^{2}}}{\imath }\) to \(\frac{1}{8{e}^{2}\imath }\).
\[{s}^{2}rx=\frac{1}{8{e}^{2}\imath }\]
Divide both sides by \({s}^{2}\).
\[rx=\frac{\frac{1}{8{e}^{2}\imath }}{{s}^{2}}\]
Simplify \(\frac{\frac{1}{8{e}^{2}\imath }}{{s}^{2}}\) to \(\frac{1}{8{e}^{2}\imath {s}^{2}}\).
\[rx=\frac{1}{8{e}^{2}\imath {s}^{2}}\]
Divide both sides by \(x\).
\[r=\frac{\frac{1}{8{e}^{2}\imath {s}^{2}}}{x}\]
Simplify \(\frac{\frac{1}{8{e}^{2}\imath {s}^{2}}}{x}\) to \(\frac{1}{8{e}^{2}\imath {s}^{2}x}\).
\[r=\frac{1}{8{e}^{2}\imath {s}^{2}x}\]
\[r=\frac{1}{8{e}^{2}\imath {s}^{2}x}\]
Collect all solutions.
\[r=\frac{1+\imath xn}{8{e}^{2}\imath {s}^{2}},\frac{1}{8{e}^{2}\imath {s}^{2}x}\]
r=(1+IM*x*n)/(8*e^2*IM*s^2),1/(8*e^2*IM*s^2*x)
