What use is there for rationalizing denominator?

[leighflix]

I've been learning algebra for the past 2 years (in high school), not once have we ever had to rationalize a denominator in a radical expression. I am now relearning Algebra and Trig., what use is there? I mean, all you're doing is switching the numerator (rational) to the Denominator (irrational) in terms of rationality. (not actually switching the numbers).

Rational / Irrational → Irrational / Rational

 

[leighflix]

After quite a while of thinking about this, the only conclusion I can come up with is to simplify it.

Say for example, 6 / √12 → 6√3 / √(12 ⋅ 3) → 6√3 / 6 → √3

 

[Mark44]

I've been learning algebra for the past 2 years (in high school), not once have we ever had to rationalize a denominator in a radical expression. I am now relearning Algebra and Trig., what use is there? I mean, all you're doing is switching the numerator (rational) to the Denominator (irrational) in terms of rationality. (not actually switching the numbers).

Rational / Irrational → Irrational / Rational

The main rationale behind this technique is that it's easier to divide an irrational number by a rational number than the other way around, especially if you don't have a calculator or computer to do the work for you.

The technique is also used for fractions that involve complex numbers. For example, $\frac 1 i$ can be simplified by multiplying by the complex conjugate over itself; i.e., by multiplying by $\frac{-i}{-i}$ (which is 1). So $\frac 1 i = \frac 1 i \cdot \frac{-i}{-i} = \frac{-i}{1} = -i$.

 

[leighflix]

Thanks, and a small side question: How do you create the fractions? (on this forum)

 

[jedishrfu]

We use Latex codes and it is rendered by the MathJax javascript library.

https://www.physicsforums.com/help/latexhelp/

There's a simple example of the quadratic formula that has a fraction in it.

 

[FactChecker]

Regarding the format of math equations: If you see an example math equation here whose format you want to mimic, you can right-click and see the Show Math As => TeX commands. Copy it and surround it with [ tex] ... [/tex]. (Note. I had to insert a space before 'tex' to stop the formatter from parsing it). Here is an example from a post above.

\frac 1 i = \frac 1 i \cdot \frac{-i}{-i} = \frac{-i}{1} = -i

 

[Mark44]

Copy it and surround it with [ tex] ... [/tex].

I never use these any more -- I find it simpler to use $$ at the beginning of the expression and $$ at the end -- it's less to type. For stuff that I want to show inline, I use $at the beginning and two more of them at the end. The$ pairs are equivalent to ....

 

[pwsnafu]

I personally consider $\frac{1}{\sqrt{2}} + \frac{1}{\sqrt[3]{3}} = \frac{\sqrt2}{2} + \frac{\sqrt[3]9}{3} = \frac{3\sqrt2+2\sqrt[3]9}{6}$ to be simpler than $\frac{1}{\sqrt2} + \frac1{\sqrt[3]3} = \frac{\sqrt2+\sqrt[3]3}{\sqrt2 \sqrt[3]3}$ but this is a YMMV thing.