Hint: if you think to take the cube of this expression, you are on the right track.

A little algebra help for those who don't remember the formula:

taking the cube of the expression in OP gives at first,

Spoiler

Now, to simplify...

So, after taking cube of the expression in question,

and simplifying, we get it (i.e., the cube) being equal to

Spoiler

Do you see something peculiar here?

Edited May 3, 20232 yr by Genady

Such math puzzles make little sense in the age of Wolfram Alpha. It solves them right away.

 

An opportunity to try handmade Latex:

[math]\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}[/math]

[math]\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}[/math]

 

Edited May 3, 20232 yr by Sensei

9 minutes ago, Sensei said:

Such math puzzles make little sense in the age of Wolfram Alpha. It solves them right away.

 

An opportunity to try handmade Latex:

7+50−−√−−−−−−−√3+7−50−−√−−−−−−−√3

 

[math]\sqrt[3]{7+\sqrt{50}}+\sqrt[3]{7-\sqrt{50}}[/math]

 

 

What do you mean, it solves them right away?

18 minutes ago, Genady said:

What do you mean, it solves them right away?

https://www.wolframalpha.com/input?i=cube+root+of(+7%2Bsqrt(50))%2Bcube+root+of(+7-sqrt(50))

 

12 minutes ago, Sensei said:

https://www.wolframalpha.com/input?i=cube+root+of(+7%2Bsqrt(50))%2Bcube+root+of(+7-sqrt(50))

 

It doesn't let me see the whole step-by-step solution, but it looks like it just keeps simplifying the original expression. If so, it misses the beautiful insight:

after the first step of taking the cube, the puzzle boils down to the simple equation,

Spoiler

x³ = 14 - 3x

which is quickly solved by inspection.