Simplifying Expressions Involving Square and Fourth Roots

[oconnk]

I was looking over a problem to make sure I hadn't messed up my arithmetic and I put the term (-SQRT(18-12SQRT(2))/6 into my calculator and it reduced it to (2SQRT(3)-SQRT(6))/6.
I found approximate values for these two expressions and they were in fact equal. So my question is, how does one figure out how to reduce expressions involving fourth roots and square roots into those involving only square roots without the use of a TI-89 or any other exact value calculator? Could you show me step-by-step on this one as an example?

 

[qntty]

Obviously the simplification happens in the numerator so I'll ignore the 1/6. To simplify this we need to take care of the difference underneath the radical. We do this by factoring. First remove any unnecessary factors.
$$-\sqrt{18-12\sqrt{2}} = -\sqrt{6(3-2\sqrt{2})}$$
You can probably see that, if this can be factored, then the factored form looks like $$(a+b\sqrt{2})^2$$. There aren't many possibilities for $$3-2\sqrt{2}$$ and by some guess and check you'll find that a=1 and b=-1.
$$-\sqrt{6(1-\sqrt{2})^2} = -\sqrt{6}(1-\sqrt{2}) = \sqrt{6} - 2\sqrt{3}$$

 

[Tobias Funke]

oconnk, you sure you're not forgetting a minus sign somewhere? They're negatives of one another.

Since 1-sqrt(2)<0, the square root of it squared (forgive the stupid wording, I'm exhausted) is sqrt(2)-1, and the rest of the above post still works.

 

[qntty]

oconnk, you sure you're not forgetting a minus sign somewhere? They're negatives of one another.

Since 1-sqrt(2)<0, the square root of it squared (forgive the stupid wording, I'm exhausted) is sqrt(2)-1, and the rest of the above post still works.

Yes and I made the same error!

 

[Anant9]

Obviously the simplification happens in the numerator so I'll ignore the 1/6. To simplify this we need to take care of the difference underneath the radical. We do this by factoring. First remove any unnecessary factors.
$$-\sqrt{18-12\sqrt{2}} = -\sqrt{6(3-2\sqrt{2})}$$
You can probably see that, if this can be factored, then the factored form looks like
(a+b2√)2
. There aren't many possibilities for
3−22√
and by some guess and check you'll find that a=1 and b=-1.
$$-\sqrt{6(1-\sqrt{2})^2} = -\sqrt{6}(1-\sqrt{2}) = \sqrt{6} - 2\sqrt{3}$$

for more on http://math.tutorvista.com/algebra/simplifying-expressions.html" log on to tutorvista.com

 

[HallsofIvy]

qntty and Anant9, don't use the tag "latex" on this form. Use "tex" or, for inline, "itex" only. I edited your posts, replacing "latex" with "tex".