Math Olympiad Question | Nice Algebra Problem | Math Olympiad Preparation

Can you solve the given rational problem involving exponents? Find the value of Rational Expression (81^x )/((81^x )+1) if (3^x )/((3^x )+1) = 2/5

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Math Olympiad Question | Nice Algebra Problem | Math Olympiad Preparation

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[anatoliy]

Simple but nice a task. Thank you so much, sir! You are a great teacher! All the best to you!

[fevengr]

Solved this one using a very similar approach. Also, when I saw that the video was only slightly over 5 minutes, I figured I might have a reasonable chance to find the correct answer. The problem yesterday was over 15 minutes and I never came remotely close to solving that one. Thanks!

[HappyFamilyOnline]

Very helpful step by step explanation👍
Thanks for sharing😊

[charlesmitchell]

Nice! I love this channel. Keeps me sharp with math so I can help my kids.

[mathsdone]

Beautiful presentation with clear explanation. Thanks sir for such a nice mathematical video. 👍👍🙏🙏🙏

[d.m.]

Solved this in less than 1 minute by taking reciprocal of the first equation and getting value of 3^x = 2/3. Then solved further by making reciprocal of the required equation.

[alster]

Solved it on my own after seeing the substitution method and as usual fast forwarded to the end to double check it matched.

[mcbeaulieu]

I love how your answers don't necessarily simplify and stay rough, that feels more "natural" to me. Keep up the great work professor 👌

[ytusr-korg]

I've learnt a lot from your channel. thank you, sir:D

[pranaykumar]

Sir also ur way of teaching is very excellent

[mykiits]

Solved it in exactly the same way as you. Thanks for video =)

[meldatv]

Another nice algebra problem that you easily solved.

[sheungmingchoi]

As LHS's denominator is 1 more than LHS's numerator, while RHS's denominator is 3 more than RHS's numerator, one could instantly divide both RHS's numerator and RHS's denominator by 3 to shrink their difference to 1 as well, i.e. (2/3)/(5/3) which is (2/3)/[1+(2/3)], so 3^x = 2/3. Almost nothing to calculate by here.

The desired is just (2/3)^4/[1+(2/3)^4]. Time it by (3^4)/(3^4) into (2^4)/(3^4+2^4), a form an average person needs mere seconds to simplify.

This is among those most lenient questions for renowned competitions.

Side note: f(a) := a/(1+a) = 1-[1/(1+a)], so f(a) is a strictly increasing function in a>0, thus 2/3 is the only possible value for 3^x. OTOH, one can see this value is unique at once by referring to the host's "linear equation" approach.

[hansschotterradler]

I got this one thanks to your teaching.

[Yt-ffhn]

Sir can you not bring the Putnam, RMO, IMO questions???
These are easy ones, but nice explanation and presentation...

[GillAgainsIsland]

Easier to just multiply numerator and denominator by 81 to get 16/97.

[paulc]

Thank you for your video. I worked out 3^X = 2/3; 81^X = 3^4X. However when I put these on I made careless mistake then wrong result.

[bentels]

I solved it almost the same way. Just the first part I did differently:

(3^x)/(1+3^x)=2/5
(1+3^x)/(3^x)=5/2
1/(3^x) + (3^x)/(3^x) = 5/2
1/(3^x) + 1 = 5/2
1/(3^x) = 3/2
3^x = 2/3

[CarlosSaucedaReyes]

I did it, And the result is the same, congratulations

[ranveeryadav]

By solving we get 3^x =2/3
So 81^x =16/27 by putting ( 16/27 )/1+16/27 then the ans is 16/97