# Thread: Having trouble with roots

1. ## Having trouble with roots

Ok, so I appreciate that this isnt a difficult question, but I am having some serious problems here...

After years of being afraid to learn maths, I have finally bitten the bullet and decided to go back to school. I want to work with sciences, so I need a strong background in maths. To prepare for going back to school (there are some paperwork issues stopping me applying immediately) I am currently re-learning using the book Algebra Demystified (I plan to work on other books in the series after im done with this one, until I am able to go to college).

Anyway, im stuck on roots, particularly simplifying roots.

I dont really understand the process, and I dont really understand WHY we simplify roots. Basically, the whole thing has gotten me a bit stumped, and i wondered if I could ask here for some help?

Thanks in advance, and im so sorry if this is a bit garbled... I will try to clarify it a bit if needed?

2. Originally Posted by skysurfer2007
True, i guess I was a bit too vague... I get that it is there to make them easier to use in a calculation, but I dont understand the process of simplifying them at all, and I cannot find an explanation as to why the answers to these problems are written the way they are etc.

I can follow the method fairly easily, but I dont understand why certain numbers end up on either side of the √ sign, and how that was reached...

For example, simplifying √27... the answer is given as 3√3³. I dont get how they got to that answer, I understand that 3*3*3= 27, but it just isnt making any sense at all to me in terms of roots... and of course, once you include variables in there too, im completely lost!
Well if you're taking something out from under the square root sign, it has to be square rooted itself. So if you simplify √27 to √9*√3, that equals 3√3.

3. Originally Posted by skysurfer2007
True, i guess I was a bit too vague... I get that it is there to make them easier to use in a calculation, but I dont understand the process of simplifying them at all, and I cannot find an explanation as to why the answers to these problems are written the way they are etc.

I can follow the method fairly easily, but I dont understand why certain numbers end up on either side of the √ sign, and how that was reached...

For example, simplifying √27... the answer is given as 3√3³. I dont get how they got to that answer, I understand that 3*3*3= 27, but it just isnt making any sense at all to me in terms of roots... and of course, once you include variables in there too, im completely lost!
That comes from the rule of roots: $\displaystyle \sqrt{(ab)} = \sqrt(a) \sqrt(b)$

In the case of $\displaystyle \sqrt(27) = \sqrt(9) \sqrt(3) = 3\sqrt(3)$ because $\displaystyle \sqrt(9) = 3$

4. Originally Posted by skysurfer2007
Ok, so I appreciate that this isnt a difficult question, but I am having some serious problems here...

After years of being afraid to learn maths, I have finally bitten the bullet and decided to go back to school. I want to work with sciences, so I need a strong background in maths. To prepare for going back to school (there are some paperwork issues stopping me applying immediately) I am currently re-learning using the book Algebra Demystified (I plan to work on other books in the series after im done with this one, until I am able to go to college).

Anyway, im stuck on roots, particularly simplifying roots.

I dont really understand the process, and I dont really understand WHY we simplify roots. Basically, the whole thing has gotten me a bit stumped, and i wondered if I could ask here for some help?

Thanks in advance, and im so sorry if this is a bit garbled... I will try to clarify it a bit if needed?
Well I'd say we simplify roots in order to make them easier to deal with in a calculation. Just like you'd say 72*4 rather than (59+13)*(7-3)

5. True, i guess I was a bit too vague... I get that it is there to make them easier to use in a calculation, but I dont understand the process of simplifying them at all, and I cannot find an explanation as to why the answers to these problems are written the way they are etc.

I can follow the method fairly easily, but I dont understand why certain numbers end up on either side of the √ sign, and how that was reached...

For example, simplifying √27... the answer is given as 3√3³. I dont get how they got to that answer, I understand that 3*3*3= 27, but it just isnt making any sense at all to me in terms of roots... and of course, once you include variables in there too, im completely lost!

6. Originally Posted by skysurfer2007
True, i guess I was a
For example, simplifying √27... the answer is given as 3√3 I dont get how they got to that answer, I understand that 3*3*3= 27, but it just isnt making any sense at all to me in terms of roots... and of course, once you include variables in there too, im completely lost!
First of all

$\displaystyle \sqrt{27} \ne$ 3√3³

It is = 3√3

I can't really understand what desn't make sense to you

$\displaystyle \sqrt{3}$ is irrational it cannot be written without roots

There is no number that I know whose square is 3

7. Originally Posted by skysurfer2007

Anyway, im stuck on roots, particularly simplifying roots.

I dont really understand the process, and I dont really understand WHY we simplify roots. Basically, the whole thing has gotten me a bit stumped, and i wondered if I could ask here for some help?

Thanks in advance, and im so sorry if this is a bit garbled... I will try to clarify it a bit if needed?
I really don't have an answer for that question, but as you learn further and further you will find that it's very useful for you.
Ok, here is the basic you might need to know...

Definition: A square root of a number is a number that when squared is equal to the given number.

example: The square root of 25 is 5 and -5

Note*: $\displaystyle 5^2=25 ~\text{and}~ (-25)^2=25. ~\text{therefore both 5 and -5 are the square roots of 25}$

$\displaystyle \sqrt4=2~and~-2$

$\displaystyle \sqrt9=3~and~-3$

Definition: $\displaystyle \sqrt a$ represent the positive square root of a number, a.
the negative square root represented by
$\displaystyle -\sqrt a$

example:
$\displaystyle \sqrt 144=12$

$\displaystyle \sqrt {\frac{9}{16}}=\frac{3}{4}$

$\displaystyle \sqrt a=a^{\frac{1}{2}}$
$\displaystyle (\sqrt a)^2=\sqrt a^2=(a^2)^{\frac{1}{2}}=a^{2\times\frac{1}{2}}=a$
$\displaystyle \therefore~(\sqrt a)^2=a$

example:

$\displaystyle (\sqrt 4)^2=4$

$\displaystyle (\sqrt 9)^2=9$

$\displaystyle (-\sqrt 9)^2=9$

Formula: $\displaystyle \sqrt a\sqrt b\Longleftrightarrow \sqrt(ab)$

example:

$\displaystyle \sqrt 2\sqrt 3=\sqrt(2\times 3)=\sqrt 6$

$\displaystyle \sqrt 3\sqrt 5=\sqrt(3\times 5)=\sqrt 15$

$\displaystyle \sqrt 12=\sqrt 4\sqrt 3=\sqrt 2\sqrt 2\sqrt3=\sqrt(2\times 2)\sqrt 3=\sqrt(2^2)\sqrt 3=2\sqrt 3$<<-----$\displaystyle \sqrt(a^2)=a$

$\displaystyle \sqrt 18=\sqrt 9\sqrt 2=\sqrt 3\sqrt 3\sqrt2=3\sqrt 2$

$\displaystyle \sqrt 28=\sqrt 7\sqrt 4=\sqrt 2\sqrt 2\sqrt7=2\sqrt 7$