Question

How do you prove this equality `sqrt(3+sqrt(3)+(10+6sqrt(3))^(2/3))=sqrt(3)+1` ?

Answer 1

See here

Explanation:

It doesn't look like they're equal, so you can't prove they are..

Answer 2

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 `3+sqrt(3)` from both ends, this becomes:

`(10+6sqrt(3))^(2/3) = 1+sqrt(3)`

Raising both sides to the power `3`, this becomes:

`(10+6sqrt(3))^2 = 1+3sqrt(3)+9+3sqrt(3) = 10+6sqrt(3)`

Note that on the left hand side we have `(10+6sqrt(3))^2` while on the right hand side we have `10+6sqrt(3)`

It seems that the exponent `2/3` should have been `1/3`.

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)` to both sides, this becomes:

`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`

Answer 3

Please see below if it is `sqrt(3+sqrt(3)+(10+6sqrt(3))^(1/3))=sqrt(3)+1`

Explanation:

It should have been `sqrt(3+sqrt(3)+(10+6sqrt(3))^(1/3))=sqrt(3)+1`

Observe that `(1+sqrt3)^3=1^3+3xx1^2xxsqrt3+3xx1xx3+(sqrt3)^3`

= `1+3sqrt3+9+3sqrt3=10+3sqrt3`

Hence `sqrt(3+sqrt(3)+(10+6sqrt(3))^(2/3))`

= `sqrt(3+sqrt3+1+sqrt3)`

= `sqrt((sqrt3)^2+2xxsqrt3xx1+1^2)`

= `sqrt((sqrt3+1)^2)`

= `sqrt3+1`