\[\left( 1.999 \right)5\text{ }=\text{ }\left( 2\text{ }\text{ }0.001 \right)5\]

Let \[x\text{ }=\text{ }2\text{ }and\text{ }x\text{ }=\text{ }-\text{ }0.001\]

Likewise, let \[y\text{ }=\text{ }x5\]

Separating the two sides w.r.t, x, we get

\[dy/dx\text{ }=\text{ }5×4\text{ }=\text{ }5\left( 2 \right)4\text{ }=\text{ }80\]

Presently, \[y\text{ }=\text{ }\left( dy/dx \right).\text{ }x\text{ }=\text{ }80.\text{ }\left( -\text{ }0.001 \right)\text{ }=\text{ }-\text{ }0.080\]

Furthermore, \[\left( 1.999 \right)5\text{ }=\text{ }y\text{ }+\text{ }y\]

\[=\text{ }x5\text{ }\text{ }0.080\text{ }=\text{ }\left( 2 \right)5\text{ }\text{ }0.080\text{ }=\text{ }32\text{ }\text{ }0.080\text{ }=\text{ }31.92\]

Hence, estimated worth of (1.999)5 is

\[31.92\]