China
Simplify ^@ \dfrac { \sqrt{ 5 } + 1 } { \sqrt{ 5 } - 1 } ^@.
Answer:
^@\dfrac { 3 + \sqrt{ 5 } } { 2 }^@
- Let us multiply the numerator and the denominator by ^@(\sqrt{ 5 } + 1).^@ @^ \begin{align} & = \dfrac { \sqrt{ 5 } + 1 } { \sqrt{ 5 } - 1 } \times \dfrac { \sqrt{ 5 } + 1 } { \sqrt{ 5 } + 1 } \\ & = \dfrac { ( \sqrt { 5 } + 1 )( \sqrt{ 5 } + 1 ) } { (\sqrt{ 5 })^2 - (1)^2 } && [a^2 - b^2 = (a + b)(a - b)] \\ & = \dfrac { (\sqrt{ 5 } + 1)^2 } { 5 - 1 } \\ & = \frac { (\sqrt{ 5 })^2 + (1)^2 + 2 \times \sqrt{ 5 } \times 1 } { 4 } && [(a + b)^2 = a^2 + b^2 + 2ab] \\ & = \dfrac { 6 + 2 \sqrt{ 5 } } { 4 } \\ & = \dfrac { 3 + \sqrt{ 5 } } { 2 }\end{align} @^
