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we are given this expression with so
many square roots and our job is to make
this ratio into a more simpler form so
can you do it first we rewrite the
square < TK of 9 as theare < TK of 3 * <
TK 3 and the square < TK of 15 as Square
< TK of 3 * Square < TK of 5 similarly
we can rewrite the square otk of 12 in
the denominator as Square < TK of 4 * <
TK of 3 which simplifies to 2 * Square <
TK of 3 now this Square < TK of 1 will
simply be equal to 1 great next we
observe that the numerator contains
terms that can be grouped the terms
Square < TK of 3 + sare < TK of 5 remain
the same while from this part we can
take Square < TK of 3 as common to get
square < TK of 3 * Square < TK 3 + < TK
5 wow the numerator factors out as 1
plus the square < TK of 3 * theare < TK
of 3 plus the square < TK of 5 for the
denominator we rearrange the terms 1 + <
TK 5 + 2 * < TK 3 as 1 + < TK of 3 + <
TK of 3 + < TK of
5 noise let us call this fraction as a
variable X now here comes the magic if
this is X then 1 /x will be this right
let us simplify it to do so we split the
fraction into two separate terms the
first term is 1 + sare < TK of 3 / this
which simplifies to 1 / < TK 3 + < TK 5
the second term is square < TK 3 + < TK
of 5 divided by this which simplifies to
1 / 1 + < TK of 3 as a next step we will
rationalize each term separately to
rationalize 1 ided by square < TK of 3 +
< TK of 5 we multiply both the numerator
and denominator by sare < TK 5 - < TK 3
use a + b * a - b = a s - b square here
to get this denominator as 5 - 3 or 2 so
1 / < TK of 3 + < TK 5 simplifies to <
TK
5- < TK 3 / 2 similarly to rationalize 1
/ 1 + < TK 3 we multiply both numerator
and denominator by square < TK of 3 - 1
again using this we get this denominator
as 3 - 1 or 2 so this simplifies to
square root of 3 -1 / 2 Now by adding
these two results since the denominators
are the same we combine the numerators
to obtain this oh look Square < TK of 3
gets cancelled out and we are left with
< TK of 5 - 1 / 2 as 1 /x so x = 2 over
the < TK of 5 -1 we will rationalize
this one last time by multiplying both
numerator and denominator by theare < TK
of 5 + 1 again using this we get the
denominator as 5 - 1 or 4 this two gets
canceled out with four and we get x = 1
+ < TK 5 / two this is giving me
goosebumps and I am in shock right now
because this number is equal to none
other than the well-known golden ratio
which is a special number that appears
in nature
art and Mathematics so all this ugly
looking square roots simplify to this
beautiful golden racio if you enjoyed
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