Simplify √(√8+√9)?

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Answer 1 · 2018-02-06T17:16:15

#sqrt(sqrt8+sqrt9)=1+sqrt2#

Let #sqrt(sqrt8+sqrt9)=sqrta+sqrtb#

Squaring #sqrt8+sqrt9=a+b+2sqrt(ab)# and as #sqrt9=3#,

we have #a+b+2sqrt(ab)=3+sqrt8=3+2sqrt2#

Hence #a+b=3# and #ab=2#

i.e. #a=3-b# and hence #(3-b)b=2# or #3b-b^2=2#

or #b^2-3b+2=0# i.e. #(b-2)(b-1)=0#

therefore #b=1# or #2#.

and then #a=2#or #a=1#

Observe that they lead to same solution

Hence #sqrt(sqrt8+sqrt9)=sqrt1+sqrt2=1+sqrt2#