Simplifying fraction with surds on both sides

[conwy]

So I'm doing an online intro to calculus course, and I've come across this problem:



They want me to simplify a surd expression which is a fraction which has SQRT(3) - 1 on top and SQRT(3) + 1 on the bottom.

So I'm a bit of a beginner here, but I figure maybe the first thing to do is try to get the surd on the top side of the fraction and 1 on the bottom. Then I can simplify the whole thing to just the top part, since any 'x' divided by 1 is simply 'x'.

There seems to be a rule (which I'm still struggling to understand) when multiplying a surd expression by itself, one side of the multiplication has to be the 'inverse' or 'opposite' of the other. I think this is referred to as difference of two squares. So this means that if I have SQRT(3) - 1 on the left side, then I should have SQRT(3) + 1 on the right side, since +1 is the 'inverse' of -1.

So I try it that way:

SQRT(3) - 1 X SQRT(3) - 1
-----------------------------------------------
SQRT(3) + 1 X SQRT(3) - 1

If I'm not mistaken, SQRT(3) X SQRT(3) = 3.
And -1 X -1 should equal just 1.
And 1 X -1 should equal -1.

So I should get:

3 + 1
------------
3 - 1

Which is really:

4
----
2

Which is really:

2

But this is nothing like the actual correct answer (highlighted)!

So I'm trying to figure out - what am I doing wrong here?

Is my error in multiplying both sides by the denominator?
Is my error in how I multiply the surds? Is a SQRT(3) X SQRT(3) not actually equal to 3?
Is my error in how I multiple the coefficients? Is 1 X -1 not actually equal to -1? Is 1 X 1 not actually equal to 1?
Is my error in how I multiple the expressions? Am I doing the multiplication as a whole in an incorrect manner?

I've been searching high and low on Google, using keywords such as: 'surd expression simplify', 'surd fraction top bottom', 'surd fraction simplify', etc, but am unable to find anything matching this problem. All the problems I find online seem to have surds on only one side of the fraction, or don't involve simplification, or are in some other way different from this problem.

I can't believe math is this hard! Maybe I'm missing something really elementary.

But honestly, having been a software developer for 15+ years, this is much much harder than software development. In software, you can Google, read manuals and eventually figure out the answer. But here I feel totally lost. There seems to be no reliable way of finding my way without just giving up and asking other people for help! If anyone can recommend some kind of book or online course that might help, it would be much appreciated.

Maybe I'm just not trying hard enough. Or maybe some things just take time and persistence.

Many thanks ahead of time for any help you can offer!

 

[MarkFL]

What you want to do here is to rationalize the denominator:

[MATH]\frac{\sqrt{3}-1}{\sqrt{3}+1}=\frac{\sqrt{3}-1}{\sqrt{3}+1}\cdot\frac{\sqrt{3}-1}{\sqrt{3}-1}=\frac{3-2\sqrt{3}+1}{3-1}=\frac{2(2-\sqrt{3})}{2}=2-\sqrt{3}[/MATH]

 

[lev888]

I am sorry, I find it hard to believe you couldn't find anything online. I just used your thread title and got a ton of matches that explain exactly what you need.
I don't see any difference between looking for answers to math and software questions. I am a software engineer.

 

[conwy]

Ok, thanks for your answer, MarkFL!

Rationalizing the denominator was exactly what I was trying to do, but it looks like I wound up with a different result than you did.

It seems that my error was in how I multiplied the top bit. So SQRT(3) - 1 X SQRT(3) - 1 = 3 - (2 X SQRT(3)) + 1.

I still don't understand how you can multiply SQRT(3) - 1 by itself and get anything other than 3 - 1. Since SQRT(3) times itself ought to be just 3, and -1 times -1 ought to be just 1.

What am I missing here?

Is there some material I need to read about how to multiply surd expressions in which the surd has a number subtracted from it?
Is there anything I could Google that would help me to understand the specific mistake I'm making here?

Thanks again for your help thusfar!

 

[conwy]

lev888, so did you find a problem online where there is a fraction in which one side has a surd minus some number and the other side has a surd plus some number? I've gotten to page 3 in the search results and simply can't find such an example.

For example, consider this search result: https://math.stackexchange.com/questions/67500/simplify-a-surd-expression.

It has surds on both sides of the fraction. It simply doesn't have a surd minus a number on one side and a surd plus a number on the other side.

Again, maybe I'm not being persistent enough. Maybe I need to go to page 6 of the search results.
Or maybe I'm missing some part of the brain that would enable me to take the solution to the above link and somehow generalize it and then apply it to my own problem?

 

[MarkFL]

Ok, thanks for your answer, MarkFL!

Rationalizing the denominator was exactly what I was trying to do, but it looks like I wound up with a different result than you did.

It seems that my error was in how I multiplied the top bit. So SQRT(3) - 1 X SQRT(3) - 1 = 3 - (2 X SQRT(3)) + 1.

I still don't understand how you can multiply SQRT(3) - 1 by itself and get anything other than 3 - 1. Since SQRT(3) times itself ought to be just 3, and -1 times -1 ought to be just 1.

What am I missing here?

Is there some material I need to read about how to multiply surd expressions in which the surd has a number subtracted from it?
Is there anything I could Google that would help me to understand the specific mistake I'm making here?

Thanks again for your help thusfar!

What you need to understand here is how to multiply binomials, what in some circles is referred to as the FOIL process.

 

[lev888]

I still don't understand how you can multiply SQRT(3) - 1 by itself and get anything other than 3 - 1. Since SQRT(3) times itself ought to be just 3, and -1 times -1 ought to be just 1.
What am I missing here?

(a - b)(a - b) = ?

 

[JeffM]

I still don't understand how you can multiply SQRT(3) - 1 by itself and get anything other than 3 - 1. Since SQRT(3) times itself ought to be just 3, and -1 times -1 ought to be just 1.

What am I missing here?

[MATH](\sqrt{3} - 1)(\sqrt{3} - 1) = \sqrt{3}(\sqrt{3} - 1) - 1(\sqrt{3} - 1) =[/MATH]
[MATH](\sqrt{3})^2 - \sqrt{3} - \sqrt{3} + 1 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} =[/MATH]
[MATH]2(2 - \sqrt{3}).[/MATH]
You probably learned that

[MATH](a - b)^2 = a^2 - 2ab + b^2 \ne a^2 - b^2 \text { unless b = 0.}[/MATH]
That doesn't change when dealing with surds: they are just numbers.

 

[conwy]

Ok, I must have been born without the math gene...

Here's my simple, naive attempt at applying the 'FOIL' principle:



1. First - I assume that refers to the first number on each side. So that should be SQRT(3) X SQRT(3). Which should be 3, right?
2. Inner - I assume that refers to the inner numbers on each side. So that should be -1 X SQRT(3). Which should be -SQRT(3), right?
3. Outer - I assume that refers to the outer numbers on each side. So that should be SQRT(3) X -1. Which is the same as the -1 X SQRT(3), which is -SQRT(3), right?
4. Last - I assume that refers to the last numbers on each side. So that should be -1 X -1. Should simply be 1, shouldn't it?

So shouldn't the result be 3 + -SQRT(3) + -SQRT(3) + 1?
And shouldn't that reduce to 3 - SQRT(3) - SQRT(3) + 1?
So shouldn't the 3 + 1 add up to 4, so we actually have 4 - SQRT(3) - SQRT(3)?
And shouldn't 4 - SQRT(3) - SQRT(3) be the same as 4 - 2(SQRT(3)), since we're subtracting the SQRT twice?

And yet markfl seems to be getting a different result: 2−√3.

Am I missing something really obvious here?

 

[conwy]

Am I missing something really obvious here?

Ok duhhh I forgot to divide it by 2!!!

Ok thanks very much everyone, I seem to have finally seen the light!

Apologies for the brash tone as well, I will try to be more calm and collected next time.

I guess the main lesson learned here: when multiplying surd expressions, apply the distributive principle.