Rationalizing the Denominator

[Luke55]

Hi, i have just a few questions on how Rationalizing the Denominator works.

First if you had to rationalise 2/radical: 5 why does it become 2 and radical 5 over 5. Why is not 10/25?

Second question is 4/3 and radical 7, why do we multiply 3 and radical 7 to get 21?

I hope i've explained it well...

 

[HallsofIvy]

Hi, i have just a few questions on how Rationalizing the Denominator works.

First if you had to rationalise 2/radical: 5 why does it become 2 and radical 5 over 5. Why is not 10/25?

The basic idea is that $\displaystyle \sqrt{a}(\sqrt{a})= (\sqrt{a})^2= a$. So to "get rid of" a square root
you multiply by that same square root, Also I hope you understand that, in order not to change the actual value of a fraction you have to multiply both numerator and denominator by the same thing.

To "get rid of" $\displaystyle \sqrt{5}$ in the denominator, multiply both numerator and denominator by $\displaystyle \sqrt{5}$. $\displaystyle \frac{2}{\sqrt{5}}= \frac{2}{\sqrt{5}}\frac{\sqrt{5}}{\sqrt{5}}= \frac{2\sqrt{5}}{(\sqrt{5})^2}= \frac{2\sqrt{5}}{5}$

I am puzzled as to why you would think it should be $\displaystyle \frac{10}{25}$! For the numerator to go from 2 to 10 you must have multiplied by 5. But multiplying the denominator by 5 does not give 25, it gives $\displaystyle 5\sqrt{5}$. In fact, it should be easy to see that $\displaystyle \frac{10}{25}= \frac{2\cdot 5}{5\cdot 5}= \frac{2}{5}$, not $\displaystyle \frac{2}{\sqrt{5}}$. Or are you simply not clear on what "$\displaystyle \sqrt{}$" means?

Second question is 4/3 and radical 7,

I think you mean $\displaystyle \frac{4}{3\sqrt{7}}$?

why do we multiply 3 and radical 7 to get 21?

We don't. We multiply both numerator and denominator by $\displaystyle \sqrt{7}$ to get
$\displaystyle \frac{4}{3\sqrt{7}}\frac{\sqrt{7}}{\sqrt{7}}= \frac{4\sqrt{7}}{3\sqrt{7}\cdot \sqrt{7}}= \frac{4\sqrt{7}}{3(\sqrt{7})^2}= \frac{4\sqrt{7}}{3\cdot 7}= \frac{4\sqrt{7}}{21}$

I hope i've explained it well...

 

[Luke55]

Thanks for the reply. I don't understand the square root sign from the looks of it. Ill find out more information. Thanks again

 

[Mr. Bland]

First if you had to rationalise 2/radical: 5 why does it become 2 and radical 5 over 5. Why is not 10/25?

The square root of 5 is in the neighborhood of 2.2. Since [MATH]5 \ne \sqrt{5}[/MATH], it is likewise true that [MATH]5 * 5 \ne 5 * \sqrt{5}[/MATH].

You can use a fraction that is the same on the top and the bottom (which is equal to 1) and multiply with it to move terms around within another fraction. Consider the following:

[MATH]\frac{2}{\sqrt{5}} = \frac{2}{\sqrt{5}} * 1[/MATH]
[MATH]\frac{2}{\sqrt{5}} = \frac{2}{\sqrt{5}} * \frac{\sqrt{5}}{\sqrt{5}}[/MATH]
[MATH]\frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{\sqrt{5}^2}[/MATH]
[MATH]\frac{2}{\sqrt{5}} = \frac{2 \sqrt{5}}{5}[/MATH]​

 

[HallsofIvy]

I don't understand the square root sign from the looks of it.

I'm not sure what you mean by this. It's pretty straight forward. $\displaystyle \sqrt{a}$ is the positive number, x, such $\displaystyle x^2= a$. The square root of 4 is 2 because $\displaystyle 2^2= 2(2)= 4$. The square root or 9 is 3 because $\displaystyle 3^2= 3(3)= 9$. The square root of 2.25 is 1.5 because $\displaystyle 1.5^2= 1.5(1.5)= 2.25$.