Why can't \sqrt{30-x^2} be simplified to \sqrt{30}-x?

[0tt0UK]

Hi all,

I am reading about trigonometry and pythagoras..

I found this (30-x²)^1/2

Would it be the same as (sqrt30 - x) ??

Is there any way to simplify it?

thanks in advance

 

[sylas]

You cannot simplify it any further. You cannot distribute the square root over the addition.

 

[0tt0UK]

thx sylas that is what I thought but I wasn't sure..

so if I had to multiply that by let's say (2+x). What would happen to the square root (1/2)?

Add the exponents due to the multiplication (1 + 1/2 = 3/2)?

 

[Mentallic]

so if I had to multiply that by let's say (2+x). What would happen to the square root (1/2)?

So you will have $$(2+x)\sqrt{30-x^2}$$ ??
Nothing happens to the square root

Oh and by the way, the rule is $$\sqrt{a^2}=a$$ for a being anything, simple or complicated. Think a bit about this rule and see if you can figure out why you can't simplify the above expression.

 

[0tt0UK]

...Think a bit about this rule and see if you can figure out why you can't simplify the above expression.

Yes I understand that, ta.

But what would be the multipication between both? That is what I don't get..

 

[Mentallic]

Think about this, $$(a+b)^2=?$$

 

[D H]

Oh and by the way, the rule is $$\sqrt{a^2}=a$$ for a being anything, simple or complicated.

The rule is $$\sqrt{a^2} = |a|$$, not $$\sqrt{a^2}=a$$, and even that rule applies for the real numbers only.

 

[0tt0UK]

Think about this, $$(a+b)^2=?$$

(a+b)^2 = (a+b)(a+b)

 

[Mentallic]

lol ok I wasn't expecting that answer. Expand it!

 

[Mentallic]

The rule is $$\sqrt{a^2} = |a|$$, not $$\sqrt{a^2}=a$$, and even that rule applies for the real numbers only.

DH let's not get picky now, are you trying to confuse him?

I am reading about trigonometry and pythagoras..

Oh and p.s. what I meant by complicated was that $a$ could be a function of anything more complicated, such as $\sqrt{30}-x$ for example
(not complex numbers)...

 

[0tt0UK]

$$(a+b)^2$$ is ok but what to do when the exponent is square root ?


That is what I don't get, the expansion!

$$(2+x)\sqrt{30-x^2}$$

 

[Mark44]

I'm with D H that $$\sqrt{x^2} = |x|$$
Consider $$\sqrt{(-3)^2} = \sqrt{9} = 3 = |-3|$$
It might be that a slight complication in pursuit of the truth can be confusing, but telling a student a rule that is incorrect is worse than confusing.

 

[sylas]

$$(a+b)^2$$ is ok but what to do when the exponent is square root ?

There's nothing you can do. (a+b)^0.5 cannot be simplified further.

That is what I don't get, the expansion!

$$(2+x)\sqrt{30-x^2}$$

That isn't an expansion. It's just writing (30-x²)^0.5 using a square root sign instead of the exponent.

Cheers -- sylas

 

[0tt0UK]

There's nothing you can do. (a+b)^0.5 cannot be simplified further.



That isn't an expansion. It's just writing (30-x²)^0.5 using a square root sign instead of the exponent.

Cheers -- sylas

thx sylas.. what would you get when multiplying (30-x²)^0.5 by (2+x) or any (a+b)?

An exponent, it would be ok. I cannot do it with the square root.

 

[sylas]

thx sylas.. what would you get when multiplying (30-x²)^0.5 by (2+x) or any (a+b)?

An exponent, it would be ok. I cannot do it with the square root.

You can't simply the multiplication either. The expressions you describe are:

$$\begin{align*} (2+x)(30-x^2)^{0.5} \\ (a+b)(30-x^2)^{0.5} \end{align*}$$​

Neither of those expressions can be simplified.

 

[0tt0UK]

...
Neither of those expressions can be simplified.

ooh.. ok! thank you all for your help

 

[Mentallic]

the rule is $$\sqrt{a^2}=a$$ for a being anything, simple or complicated. Think a bit about this rule and see if you can figure out why you can't simplify the above expression {$$\sqrt{30-x^2}$$}

Think about this, $$(a+b)^2=?$$

I was trying to give 0tt0UK an understanding of why this cannot be simplified to $$\sqrt{30}-x$$ !

The rule is $$\sqrt{a^2} = |a|$$, not $$\sqrt{a^2}=a$$, and even that rule applies for the real numbers only.

I'm with D H that $$\sqrt{x^2} = |x|$$
Consider $$\sqrt{(-3)^2} = \sqrt{9} = 3 = |-3|$$
It might be that a slight complication in pursuit of the truth can be confusing, but telling a student a rule that is incorrect is worse than confusing.

Seriously... have you guys not figured out yet what 0tt0UK's mathematical level is? He's learning about pythagoras' theorem for Christ sake! What is the point of saying "oh, but $$\sqrt{a^2}=|a|$$ since... wait, you don't understand why this is true? *goes off on a massive tangent to explain something that is unnecessary for 0tt0UK's case*... but really, this won't be happening in Pythagoras' theorem because side lengths in triangles are positive... not negative, and not complex".
He has no need to learn the refined rule. It's just the same reason why a teacher would never even dare to explain to their students why such a rule doesn't work for complex numbers, since they've never heard about them before, and it would drag the lesson in the wrong direction. 0tt0UK can come back to the proper rule when necessary. Not now.

$$(a+b)^2$$ is ok but what to do when the exponent is square root ?

ooh.. ok! thank you all for your help

That's not what I was trying to get at
I'm trying to shed some light on why $$\sqrt{30-x^2}\neq \sqrt{30}-x$$

Think about this, in the rule $$\sqrt{a^2}=a$$ (where a is a length... thus, positive)
what about if we substituted a=x+y ? Then we'll have $$\sqrt{(x+y)^2}=x+y$$

But expand the $$(x+y)^2$$ and see if you notice something. Now do the same for $$(\sqrt{30}-x)^2$$ and see what you get.