How to Rationalize the Denominator of a Complex Fraction?

[Trail_Builder]

not sure how to solve this. havnt been told, but would like to know, thnx

Homework Statement

rationalise the denominator of

$$\frac{1}{\surd2 + \surd3 + \surd5}$$

Homework Equations

The Attempt at a Solution

i only know how to rationalise it if its like only $$\frac{1}{\surd2 + \surd3}$$



hope you can help

 

[Astronuc]

Start with

$$\frac{1}{(\sqrt2 + \sqrt3) + \sqrt5}*\frac{(\sqrt2 + \sqrt3) - \sqrt5}{(\sqrt2 + \sqrt3) - \sqrt5}$$

 

[Trail_Builder]

o rite thanks, just what i was looking for :D

 

[Trail_Builder]

hmmm, the answer book has a different answer to me.

ill show my working, and can you confirm I've done it right. thanks.

$$\frac{1}{(\sqrt2 + \sqrt3) + \sqrt5}*\frac{1}{(\sqrt2 + \sqrt3) - \sqrt5}$$

$$\frac{(\surd2 + \surd3) - \surd5}{(\surd2 + \surd3)^2 - 5}$$

$$\frac{-\surd5}{-5\surd2 -5\surd3}$$

$$\frac{-\surd5(-5\surd2 + 5\surd3)}{(-5\surd2 -5\surd3)(-5\surd2 + 5\surd3)}$$

$$\frac{5\surd10 - 5\surd15}{50 - 75}$$

$$\frac{\surd10 - \surd15}{10 - 15}$$


$$\frac{\surd10 - \surd15}{-5}$$

right now i think that working is correct? right?

but the answer book gives

$$\frac{2\surd3 + 3\surd2 - \surd30}{12}$$

:S are they the same? or different? and/or why?

hope you can clear it up :D

p.s. dang that was tedious to tex all that hehe.

 

[Astronuc]

I corrected my earlier response. Initially I had focused on the denominator, but the numerator must equal the denominator in the second term in order to preserve the value of the initial expression.

$$\frac{(\surd2 + \surd3) - \surd5}{(\surd2 + \surd3)^2 - 5}$$ is correct.

Now looking at the denominator

$$(\sqrt2 + \sqrt3)^2 - 5$$ = $$(2+3+2\sqrt2\sqrt3) - 5$$

which is just $$2\sqrt2\sqrt3$$

The multiply the full expression by $$\frac{\sqrt6}{\sqrt6}$$

 

[Trail_Builder]

o rite yeah soz, i actually accounted for that in the first one without texing it, hehe

can you please check my workings

thanks

 

[Astronuc]

$$\frac{-\surd5}{-5\surd2 -5\surd3}$$ This part is not correct. I'm not sure how one manage to get this.

See my previous post regarding the denominator.

 

[Trail_Builder]

wooops. so how would i go from the step before to the next stage?

 

[Astronuc]

Starting with $$\frac{(\surd2 + \surd3) - \surd5}{(\surd2 + \surd3)^2 - 5}$$

take what I did with the denominator, which gives

$$\frac{(\surd2 + \surd3) - \surd5}{(2+3+2\sqrt2\sqrt3) - 5}$$

= $$\frac{(\surd2 + \surd3) - \surd5}{2\sqrt2\sqrt3}$$

and you can take it from there.

 

[Trail_Builder]

thanks :D all sorted