# Thread: is square root of 1 equal 1?

1. ## is square root of 1 equal 1?

1) Is $\sqrt{1} = 1$ ?

2) And if so, then $1 + \sqrt{x} = \sqrt{1} + \sqrt{x} = \sqrt{1 + x}$ ?

3) If 1 and 2 is true, then:

$(1 + \sqrt{x})^2 = (\sqrt{1 + x})^2$

But if you actually square $(1 + \sqrt{x})^2$ and $(\sqrt{1 + x})^2$, you'll find out that they are different:

$(1 + \sqrt{x})^2 = (1 + \sqrt{x})(1 + \sqrt{x}) = 1 + 2\sqrt{x} + x$

$(\sqrt{1 + x})^2 = 1 + x$

Why is that?

2. $\sqrt{a+b}=\sqrt a+\sqrt b$ is false, but it's true that $\sqrt{a+b}\le\sqrt a+\sqrt b.$

3. That's intresting information. Maybe some links on the topic, where I can have some in-depth study of the situation (c) (all the rights to Jersey Shore) here? Youtube videos would be great, but some text explanation will do too.

4. Originally Posted by Aero763
That's intresting information. Maybe some links on the topic, where I can have some in-depth study of the situation (c) (all the rights to Jersey Shore) here? Youtube videos would be great, but some text explanation will do too.
Your original post is essentially a proof that $\sqrt{a} + \sqrt{b} \ne \sqrt{a + b}$ for arbitrary $a, b$. In fact, from the last two lines you already have the inequality that Krizalid posted if you replace $1$ with $\sqrt{y}$.

5. Thank you guys.

I got which is the problem step was.

But what is the difference between $\sqrt{1} + \sqrt{x}$ and $\sqrt{1 + x}
$
?

I want to get some intuition: why it is different?

Because on the first glance they not that different: you can say they look basically the same.

So, let me get it straight:

What i got from this discussion - is that two numbers under one root - is not two numbers, but one.

Thank you, gentlemen.

6. Originally Posted by Aero763
But what is the difference between $\sqrt{1} + \sqrt{x}$ and $\sqrt{1 + x}
$

I want to get some intuition: why it is different?
Plug $x = 4: \;\;\;\;\;\;\;\;\; 1+\sqrt{4} = 1+2 = 3$ and $\sqrt{1+4} = \sqrt{5}$.

7. Originally Posted by Aero763
2) And if so, then $1 + \sqrt{x} = \sqrt{1} + \sqrt{x} = \sqrt{1 + x}$ ?
Is 1 + sqrt(4) = sqrt(5)?

8. Originally Posted by Aero763
Because on the first glance they not that different: you can say they look basically the same.
I don't understand this statement. I suppose $\frac{1}{a}+\frac{1}{b}$ looks "basically the same" as $\frac{1}{a+b}$?

9. 2) is wrong.
therefore, the next statements are wrong.