Japanese | Math Olympiad | A Nice Algebra Problem
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- Опубліковано 17 чер 2024
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Find the value of x?
How to solve x√4√2√2=32
In this video, we'll show you How to Solve Math Olympiad Question A Nice Exponential Equation x√4√2√2=32 in a clear , fast and easy way. Whether you are a student learning basics or a professtional looking to improve your skills, this video is for you. By the end of this video, you'll have a solid understanding of how to solve math olympiad exponential equations and be able to apply these skills to a variety of problems.
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Why going through all these elevations/simplification? The lhs writes: (2^2*(2*2^1/2)^1/2)^1^x. Starting from the inside and expanding, step-by-step, toward the outside parenthesis using basic power formulas (to simplify notations, I wrote a^b/c instead of a^(b/c) as it should have been), you get:
2 * 2^1/2 = 2^(1+1/2) = 2^3/2
(2^3/2)^1/2 = 2^(3/2*1/2) = 2^3/4
2^2 * 2^3/4 = 2^(2+3/4) = 2^(8/4+3/4) = 2^11/4
(2^11/4)^1/x = 2^11/4x (where 11/4x is to be read as 11/(4x) )
In other words, the original equation can be rewritten as 2^11/4x = 2^5. As exponentiation is injective, this is equivalent to 11/4x = 5 and, thus, 11 = 20 x, i.e. x=11/20.
This approach is of course equivalent to yours but, in my opinion (I'm open to contradiction'^^) is "simpler" in the sense that it only involves applying formulas minimizing the reasoning of the form "elevate both parts to ..." that (seems to me) tend to be confusing to "beginners"
P.S. The "As exponentiation is injective" argument is there only for the sake of being rigorous in my post's argumentation and should be ignored by whoever doesn't know what an "injection" is and considers that thee equivalence 2^11/4x = 2^5 11/4x = 5 obvious ;)
Hum .... WITH ALL DUE RESPECT FOR YOUR WORK, OF COURSE! I am aware of how time consuming it is to set up this kind of contents. So, thx for your useful work and passion, mate
Very nice solution genius, I appreciate your efforts you have done ❤
Thanks and welcome ❤
👍, but...
"Where 2^11/4x is to be read as 2^(11/(4x))" !!! 🥰
@@rabotaakk-nw9nm thx for appreciating this kind of details ... I'm gonna screenshot your reaction to show my students that I do not only aim at being a pain in the ass when insisting that they care a bit about being consistent in their notations ;)
Wish you good luck.....keep uploading this type of content 🎉
Sure 😊 I will try my best ❤
!
??
x= 11/20
Yes, You are right, I appreciate that ❤
x=11/20
Yes, you have done the right! I appreciate that ❤