# 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

Observe that

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