# Why does sqrt(24) = 2*sqrt(6)?

1. Dec 18, 2011

I realize the process for simplifying square roots is to break it down into lowest factors and then bring the pairs out of the radical, but why does that work? Can anybody offer an explanation?

I know it's a really elementary question but i never understood why it works!

2. Dec 18, 2011

### iRaid

$\sqrt{24}$=$\sqrt{(4)(6)}$
Square root of 4? = 2...
$2\sqrt{6}$=$\sqrt{24}$

3. Dec 18, 2011

ya i get that, but i don't understand why it works. To me it's just a little trick that works but i don't know why

4. Dec 18, 2011

### Deveno

well, it all hinges on this fact:

for a,b ≥ 0, √(ab) = (√a)(√b).

how can we prove this? well, we can just square (√a)(√b) and see if we wind up with ab.

to do THAT, we need another fact (which is probably easier for you to swallow):

(xy)2 = x2y2

because: (xy)2 = (xy)(xy) = x(yx)y = x(xy)y = (xx)(yy) = x2y2.

so...[(√a)(√b)]2 = (√a)2(√b)2 = ab

(by the definition of square root), and since a,b ≥ 0, √a, √b ≥ 0 (we are only interested in the POSITIVE square root), and positive (to be truly accurate, non-negative) times positive is positive (if one of a,b (or both) is 0, then ab = 0, and the square root of 0 is 0).

therefore, (√a)(√b) is indeed the positive square root of ab.

5. Dec 18, 2011

thanks that was great! But can you explain how you did...

(xy)(xy) = x(yx)y

?

what is that step? I knew (xy)(xy) = x^2y^2 but i don't get your intermediate steps... The rest was crystal clear thank you

6. Dec 18, 2011

### iRaid

x(yx)y is just multiplying, you can multiply in any order and you will always get the same number:

Example: 2*3*4*5 = 5*3*4*2 = 4*5*3*2 = etc..

7. Dec 18, 2011

### mathwonk

square both sides. you get the same thing. then what?

8. Dec 18, 2011

oh that's true. Thanks for the help iRaid

24=4*6... so they're equal lol

thanks everyone

9. Dec 19, 2011

### Deveno

in more complete detail:

(xy)(xy) = ((xy)x)y = (x(yx))y = x(yx)y

the re-positioning of the parentheses is justified by the associative law for multiplication:

for any 3 (real, rational, integer) numbers a,b,c:

(ab)c = a(bc).

associativity is often over-looked as a property of multiplication because we just take it for granted and often write:

abc, for any triple (or longer) product.

so i could have just written:

(xy)2= x*y*x*y = x*x*y*y = x2y2

(where all i did is "switch" the middle two factors, since multiplication for (real, rational, integer) numbers is also commutatitive).