(3+√5)^-3 + (3-√5)^-3=?

1 Answer
Tazwar Sikder
Mar 15, 2018

#frac(9)(4)#

Explanation:

We have: #(3 + sqrt(5))^(-3) + (3 - sqrt(5))^(-3)#

#= frac(1)((3 + sqrt(5))^(3)) + frac(1)((3 - sqrt(5))^(3))#

Let's combine the fractions:

#= frac((3 - sqrt(5))^(3) + (3 + sqrt(5))^(3))((3 + sqrt(5))^(3)(3 - sqrt(5))^(3))#

#= frac((3^(3) + 3 cdot (3)^(2) cdot - sqrt(5) + 3 cdot 3 cdot (- sqrt(5))^(2) + (- sqrt(5))^(3)) + (3^(3) + 3 cdot (3)^(2) cdot sqrt(5) + 3 cdot 3 cdot (sqrt(5))^(2) + (sqrt(5))^(3)))(((3 + sqrt(5))(3 - sqrt(5)))^(3))#

We can apply the algebraic identity #(a + b)(a - b) = a^(2) - b^(2)# to the denominator:

#= frac((27 - 27 sqrt(5) + 45 - 5 sqrt(5)) + (27 + 27 sqrt(5) + 45 + 5 sqrt(5)))((3^(2) - (sqrt(5))^(2))^(3))#

#= frac(54 + 90)((9 - 5)^(3))#

#= frac(144)(4^(3))#

#= frac(144)(64)#

#= frac(9)(4)#

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