- #1

Astro

- 48

- 1

- Homework Statement
- (Teaching myself and asking for help.)

Is 5^(2/2) positive, negative, or both?

- Relevant Equations
- a^(n/b)

(If I should have posted this in the Math thread instead of the Homework thread, please let me know.)

I have

I've typed my questions and solutions attempts below.

In an unrelated question (Q4): Can I

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I have

**three**questions which I will ask in sequence. They all relate to each other.I've typed my questions and solutions attempts below.

**I've also attached a hand-written version of this post (see bottom)**since, to be honest, I much prefer scanning my work than typing it out (easier to format by hand and much faster to post).**Q1)**Is the ##(\sqrt 5)^2## +ve, -ve, or ±?There is ambiguity because the answer is different depending on the method of solution. (Obviously I'm unaware of some important math rules.) The examples below illustrate the problem.

__EG__.

Use order of operations BED MAS rule

(BED MAS: Brackets > Exponents > Division/Multiplication > Addition/Subtraction)

which states that we resolve expressions within brackets first.

__SOLUTION #1__:

(√5)^2 = (± 2.2360...)^2 =

**+**5

(Note: Not sure how to get equal signs to align in Latex on separate lines. :-{ Tried the \begin{align} stuff but didn't work. The latex coding seems very buggy on this website :S )

__However__, there is a power rule that states

(EQUATION #1)

## a^{m/n} = \sqrt[n] {a^m} = (\sqrt[n] a)^m##

when a ≥ 0 ; n,m∈I|n≥2 .

__SOLUTION #2__:

∴ ## (\sqrt 5)^2 = \sqrt {5^2} = \sqrt {25} = \pm 5 ##

since radicals with an even index have both +ve and -ve roots.

EG. ● -5 X -5 = 25

● 5 X 5 = 25

∴ √ 25 =

**±**5

-5 is obviously not equal to +5. However, if EQUATION#1 is true, then SOLUTION#1 must be ± as well!

__Right__?!

If true, that would mean that

## (\sqrt 5)^2 = ±5 \neq (\pm 2.2360...)^2 ##

Instead,

(√5)^2 =

**±**(2.2360...)^2 = ±5__Right__?

So that means that when

*n*AND*m*are even, that when solving, a ± must be added, regardless of which order ## a^{m/n}## is resolved in.__Right__?

So to reiterate, my question is: What is the mathematically correct way to solve a radical of intermediate form ## a^{m/n}## in this kind of case?

**Q2)**Also, the decision of whether to add a ± must be made__before__simplifying the exponent.__Right__??EG.

## (\sqrt 5)^2 = (5^{1/2})^2##

continued: ## = 5^{2/2}##

continued: =

**±**5^1 (Notice how 2/2 reduced to 1.)

Because if we simplified 1st then we'd get

## 5^{2/2} = 5^1 = +5 ##

VERSES 5^{2/2} = ±5

EG 2.

We know that ## \sqrt {25} = ± 5 ##

But that's only true if ## \sqrt {25} = 25^{1/2} = (5^2)^{1/2} = 5^{2/2} = ±5^1 = ±5##

**Q3) In summary, for questions #1 and #2:**What is the correct way of solving the types of cases mathematically? I realize that normally people only concern themselves with the principal root (ie. positive root) but the problem is that there's a lack of consistency if rules are haphazardly applied. I'm assuming that there are standardized rules that cover these types of scenarios but I can't seem to find any in textbooks or online. (When searching online, you get mostly elementary level explanations that focus mostly in the positive roots and mostly avoid tackling these special scenarios as I have exemplified. )Any help with these kinds of questions would be appreciated.------------------In an unrelated question (Q4): Can I

*please please please*just scan my work as images and post them? It is seriously awkward and way too time consuming to type it up all using half-working latex script. It makes me want to avoid this website (which otherwise is quite good). (Sadly, other free websites seem to want you to type to...grumble grumble. >_< ) Any suggestions for websites where you can ask questions by posting scanned written work instead of typing everything? Or at least, can you*please*include a fully operating user-friendly symbol math editor that I can use with this textbox? It took me over two hours to type this when I could have written it up in less than 15min. I*did*use your Latex guide (https://www.physicsforums.com/help/latexhelp/) for coding--but Latex here wasn't working well. (I.e. Really buggy or not working at all.) Even when some Latex worked, it then stopped working and I had to delete and retype before it worked again when previewing in Chrome web browser. And both times I typed the same code identically.----------------------

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