How do you prove this equality #sqrt(3+sqrt(3)+(10+6sqrt(3))^(2/3))=sqrt(3)+1# ?
3 Answers
See here
Explanation:
It doesn't look like they're equal, so you can't prove they are..
The given equation is false, but we can prove:
#sqrt(3+sqrt(3)+(10+6sqrt(3))^color(red)(1/3)) = sqrt(3)+1#
Explanation:
Given to prove:
#sqrt(3+sqrt(3)+(10+6sqrt(3))^(2/3)) = sqrt(3)+1#
We can perform a sequence of reversible steps until we reach a simplified equation that is known to be true.
Noting that both sides of the equation to be proved are positive, it is reversible to square both sides to get:
#3+sqrt(3)+(10+6sqrt(3))^(2/3) = 3+2sqrt(3)+1 = 4+2sqrt(3)#
Subtracting
#(10+6sqrt(3))^(2/3) = 1+sqrt(3)#
Raising both sides to the power
#(10+6sqrt(3))^2 = 1+3sqrt(3)+9+3sqrt(3) = 10+6sqrt(3)#
Note that on the left hand side we have
It seems that the exponent
Then we could write a proof as follows:
#10+6sqrt(3) = 1+3sqrt(3)+9+3sqrt(3)#
#color(white)(10+6sqrt(3)) = 1+3(sqrt(3))+3(sqrt(3)^2+(sqrt(3))^3#
#color(white)(10+6sqrt(3)) = (1+sqrt(3))^3#
Taking the cube root of both ends, we find:
#(10+6sqrt(3))^(1/3) = 1+sqrt(3)#
Adding
#3+sqrt(3)+(10+6sqrt(3))^(1/3) = 3+2sqrt(3)+1#
#3+sqrt(3)+(10+6sqrt(3))^(1/3) = (sqrt(3)+1)^2#
Then taking the square root of both sides, we find:
#sqrt(3+sqrt(3)+(10+6sqrt(3))^(1/3)) = abs(sqrt(3)+1) = sqrt(3)+1#
Please see below if it is
Explanation:
It should have been
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