Line integral - confusion on squares and square root terms

[robertjford80]

line integral -- confusion on squares and square root terms

Homework Statement

Do you see where they have sqrt(16 sin^2t etc = 5? How do they get that, the answer should be 7, the square root of 16 is 4, sin^2 + cos^2 is 1 and the square root of 9 is 3, 3 + 4 = 7. It's like they're taking the square root of 16 twice.

 

[algebrat]

$3+4=7$

$3^2+4^2=5^2$

 

[algebrat]

Powers are to multiplication as multiplication is to addition, but not power is to addition.

multiplication/division distributes across addition/subtraction
powers/roots distribute across multiplication/division

$a(b+c)=ab+ac$

$(ab)^c=a^cb^c$

$(a+b)^c\ne a^c+b^c$, for most values of $a,b,c\in$ℂ

However, in a field of characteritic p, $(a+b)^p=a^p+b^p$, (or something like that, idk) which every grad student snickers and calls this the freshman's dream, because countless freshman use this rule incorrectly.

In your example, one might think you tried to do the following:

$(16+9)^{1/2}=16^{1/2}+9^{1/2}=4+3=7.$

 

[robertjford80]

I still don't get it.

sqrt(16 sin^2 + 16 cos^2 + 9) =

sqrt[16 (sin^2 + cos^2) + 9] =

sqrt(16(1) + 9) =

sqrt(16) + sqrt(9) =

4 + 3 = 7

 

[Infinitum]

sqrt(16(1) + 9) =

sqrt(16) + sqrt(9) =

This is an illegal step.

Just remolding what Algebrat posted, you cannot have $(a+b)^n = a^n + b^n$, for most real values of a,b,n. Meaning, you cannot just split away the square root into two terms. One way you can see why this is incorrect is if you know the binomial theorem for a non-integral index.

It actually goes like,

$$\sqrt{16 + 9} = \sqrt{25}$$

 

[algebrat]

Yes, last line of infinitum's post fits in nicely to your last post robertjford80.

One more check, robertjford80, is the following true?

$\sqrt2=\sqrt{1+1}=\sqrt1+\sqrt1=1+1=2?$

What does it tell us that is definitely wrong? What did I do that I shouldn't have?
The moral is, it is hard to know in math when a rule that seems like it might be true, is actually false. This is a huge theme in math. Constructing counterexamples is a big thing in math. We are finding counterexamples to a rule that is usually false. It is true in some cases, for instance the prime field thing, or if we are adding zero (if and only if?).

 

[robertjford80]

Ok, I get it now