Algebra of irrational numbers

Question

Why is $$\sqrt{7 + 2\sqrt{10}} = \sqrt2 + \sqrt5 $$

I can't seem to prove it so can someone help me out in doing so if it is possible.

And if it can't be proven, is there and explanation to why this is so?

Two positive numbers with the same square are the same number. — Jack D'Aurizio, Jun 09 '17 at 10:36

score 4 · Accepted Answer · answered Jun 09 '17 at 08:16

4

Hint. Note that $7 + 2\sqrt{10}=2+2\sqrt{2}\sqrt{5}+5$.

answered Jun 09 '17 at 08:16

Robert Z

147,345

score 1 · Answer 2 · answered Jun 09 '17 at 08:18

1

Hint: What happens when you square both sides of the equation? Positive numbers have $2$ (real) square roots, how can you distinguish between the $2$?

answered Jun 09 '17 at 08:18

Zestylemonzi

4,205

score 0 · Answer 3 · answered Jun 09 '17 at 08:18

0

Hint: One way to verify a (proposed) square root is to square the answer.

answered Jun 09 '17 at 08:18

paw88789

41,907

Algebra of irrational numbers

3 Answers3