MHB Can this Simplification Problem be Solved Using Elementary Methods?

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The discussion revolves around simplifying the expression 2 divided by the square root of a complex polynomial involving the fourth root of 5. Participants explore whether this simplification can be achieved using elementary methods. A key insight involves substituting u for the fourth root of 5 and recognizing a relationship between polynomials. The solution ultimately reveals that the expression simplifies to 1 plus the fourth root of 5. The conversation also highlights the importance of careful manipulation and observation in solving such problems.
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Problem:
Simplify $ \dfrac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}$.

I just can't see a way to solve it...

I hope someone could give me some hints if this problem could be solved using only elementary methods.

Thanks in advance.
 
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Let's let:

$\dfrac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=x$

Let also:

$u=\sqrt[4]{5}$

and we have:

$\dfrac{2}{\sqrt{4-3u+2u^2-u^3}}=x$

$2=x\sqrt{4-3u+2u^2-u^3}$

$4=x^2(4-3u+2u^2-u^3)$

Now, if we observe that:

$-u^5+5u+4=(1+u)^2(4-3u+2u^2-u^3)$

Then we may state that:

$\displaystyle x=\dfrac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=1+\sqrt[4]{5}$
 
Last edited:
MarkFL said:
Let's let:

$\dfrac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=x$

Let also:

$u=\sqrt[4]{5}$

and we have:

$\dfrac{2}{\sqrt{4-3u+2u^2-u^3}}=x$

$2=x\sqrt{4-3u+2u^2-u^3}$

$4=x^2(4-3u+2u^2-u^3)$

Now, if we observe that:

$-u^5+5u+4=(1+u)^2(4-3u+2u^2-u^3)$

Then we may state that:

$\displaystyle x=\dfrac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=1+\sqrt[4]{5}$

Bravo and thanks, Mark!(Smile) Admittedly, I would have to think for a short while before I realized $-u^5+5u=0$.:o Hehehe...
 
Last edited:
Hey anemone! :)

anemone said:
Problem:
Simplify $ \dfrac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}$.

I just can't see a way to solve it...

I hope someone could give me some hints if this problem could be solved using only elementary methods.

Thanks in advance.

To simplify such an expression, the trick is to multiply numerator and denominator by something smart.
Now what kind of smart thing can we come up with?
Let's pick $(1+\sqrt[4]{5})$ or if it does not work we can next try $(1-\sqrt[4]{5})$.

That brings:

$\dfrac{2(1+\sqrt[4]{5})}{\sqrt{(1+\sqrt[4]{5})^2(4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125})}}$

It's a bit of work to work that out, but then we'll get:

$\dfrac{2(1+\sqrt[4]{5})}{\sqrt{4 + 5\sqrt[4]{5}-(\sqrt[4]{5})^5}} = \dfrac{2(1+\sqrt[4]{5})}{\sqrt{4 + 5\sqrt[4]{5}-5\sqrt[4]{5}}} = 1+\sqrt[4]{5}$@Mark: Btw, can you replace your $-u^5 + 5u^4 + 4$ by $-u^5 + 5u + 4$, since otherwise it won't come out as 4?
 
I should add, that the "observation" I spoke of might come about by using division to find:

$\displaystyle \frac{-u^5+5u+4}{4-3u+2u^2-u^3}=(1+u)^2$

I like Serena is correct (thank you), I did have a typo in my first post, which I will correct now.
 
I like Serena said:
Hey anemone! :)
To simplify such an expression, the trick is to multiply numerator and denominator by something smart.
Now what kind of smart thing can we come up with?
Let's pick $(1+\sqrt[4]{5})$ or if it does not work we can next try $(1-\sqrt[4]{5})$.
Thanks, ILikeSerena...that is an useful and handy hint to me!
 
MarkFL said:
I should add, that the "observation" I spoke of might come about by using division to find:

$\displaystyle \frac{-u^5+5u+4}{4-3u+2u^2-u^3}=(1+u)^2$

ILikeSerena is correct (thank you), I did have a type in my first post, which I will correct now.

anemone said:
Thanks, ILikeSerena...that is an useful and handy hint to me!

It must be nice that you can fix typos in quotes.
I'm always afraid to make typos, since if someone quotes me I cannot correct it anymore.
And hey, you two just made another typo! ;)
 
(Rofl) I am the typo king! (Tmi)
 
MarkFL said:
(Rofl) I am the typo king! (Tmi)

I would have been more amused if you made a type in that declaration (that was intentional, by the way). XD
 
  • #10
Chris L T521 said:
I would have been more amused if you made a type in that declaration (that was intentional, by the way). XD

I did miss a golden opportunity there! (Smirk)
 

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