An Ingenious Trick for Evaluating a Tricky Algebraic Expression a^8+7/a^4 | Math Challenge

This math Olympiad tutorial video teaches you a cool trick / method for finding the value of the algebraic expression (sum of an octic polynomial and the reciprocal of a quartic polynomial) a^8+7/a^4 given the value of a-1/a using perfect square formula and substitution.

Math challenge
Math contest
Mathematics contest
Math competition
Mathematics competition
Math Olympiad
Mathematical Olympiads
International Mathematical Olympiad (IMO)
American Mathematics Competition (AMC)
Caribou Math Contests
American Regions Mathematics League (ARML)
Math league
Mathcounts
MathCON
SAT
ACT
Euclid
Fermat
Cayley
Pascal
Kangaroo
Canadian Math Kangaroo Contest
Canadian Intermediate Math Contests
Canadian Senior Math Contests
National Mathematics Talent Contests (NMTC)
Bhaskaracharya Mathematics Talent Search Competition.
Madhava Mathematics Competition (MMC)
EE, NDA, BITSAT, MHCET
Math
Algebra
Calculus
Geometry
Trigonometry
Arithmetic
Equation
Solution
Linear equation
Quadratic equation
Cubic equation
Algebraic equation
Exponential equation
Logarithmic equation
Radical Equation
System of equations
Quadratic formula
Perfect square formula
Factor
blackpenredpen
premath
sybermath
Math booster
Nancy
Organic chemistry tutor
Khan
Neha Agrawal Mathematically Inclined
MindYourDecisions
just algebra
Mathologer
Polynomial
Radical
Fraction
Rational function

[icems.a.]

Mientras mas videos veo, de este canal, mas admiro el talento de su productor

[owlsmath]

Wow! Wasn’t expecting the substitution for 7. That was a good move

[piman]

Wow, some Grand Master chess algebra there - loved it!

[sy]

Today's problem is interesting, because, after getting a^8+7/(a^4), it is transformed to a^8+(a^4+1/(a^4))/(a^4). Thank you for explaining.
If you want to treat other problems, I would like to inform as below:

[information]
I am also creating and uploading mathematics videos like you, so I would like to introduce you. 　
In my case, with regard to mathematics, I had a time to study math for about only 1 to 2 years at university.
When I entered university, I passed the faculty (that students can study normal math) of a national university,
but my parents did not admit me to go to the university.
It is a bitter experience. Therefore, after reaching retirement age, I started creating and uploading mathematics videos.
However, I realized that there were some mistakes about using math words when I explain. (>_<)
Some of the corrections were not made in time, so they are uploaded as they are.
(Please accept this situation.)

Nowadays, I am planning as follows:
"I will publish only my own problems at first, and after various mathematics YouTubers introducing the methods in their videos,
I will upload a video of the answers that I have prepared. So, mathematics YouTubers can compete each other for how to solve.”

That's why I wrote this in this comment.
There are 40 questions in total (excluding the Appendix). [ Note: Most of them are integer problems. ]
I prepared English versions and French versions for them. (But I am a Japanese. So Japanese is used in my videos for the oral explanation.)
For these problems, please refer to the following video.
[Video about 40 problems (Daily motion)]
SY_Math-Science_045 ( [Extra edition] The Special Event - 3 Second half) - Videos Dailymotion
[All my past videos (the 44th and 55th of these are related to the math problems)]
[Daily motion] → SY math science videos - Dailymotion

There are many number problems in the range of high school 10th (and 11th) grade and junior high school.
The problems were created based on the concept of "There are more than one solutions!"
Some of them are similar to some university entrance exams, and some of them are mathematical puzzles.
In addition, there are problems that can be solved by "the Try and Error method."
As for some problems, calculations are troublesome, concerning them, using calculator is admitted.
I guess you will be interested in some of them (or just a few?) if you like math.
If you find my problems interesting, please include them on your videos.
Instead, if you treat the problems that I made, please introduce the source of the problem and advertise my videos because I want to spread them.
(I need the propaganda.)

By the way, regarding video creation, it corresponds to question selection/creation, recording, BGM selection/input, and undercard puppet show,
editing (various cuts, text input, etc.), thumbnail creation/implementation, video management, etc. I do everything by myself.
Therefore, it takes more than a week from start creating to uploading per one video.
Therefore, I cannot make and upload my videos about something with related to them soon. (It takes time.)

"I apologize for writing so much information." Thank you very much for reading my comment.
(I look forward to your future success.) (^o^)

[Appendix]
Of the 40 problems (that I mentioned above), I introduce only 7 problems as samples below.
(Calculators are not admitted to use.) [The last one is not an integer problem.]

Q2-(2)
p^4 - 20p^3 + 90p^2 + 20p - 86 = q (p, q: prime number)　　
Find (p, q)

Q3-(8)
m^4 + 4m^2 + 8m + 9 = p^2 (m: integer, p: prime number)　
Find (m, p)

Q4-(3)
p^8 - 16q^8 = rst (p, q, r, s, t: prime number)
Find (p, q, r, s, t)

Q4-(5)
2m^3 + 4m^2n + 5m^2 + 4mn + 4n^2 + 2m + 4n = 2024 (m, n: integer)
Find (m, n)

Q4-(10)
a+b+c+d+e = a^3+b^3+c^3+d^3+e^3 = 5 (a, b, c, d, e: integer) (a≧b≧c≧d≧e)
Find more than 6 solutions.　　　　
(If there are 6 or less than 6 solutions, prove it.)

Q5-(4)
2^a + 3^b + 4^c + 6^d = n! (a, b, c, d: integer, n: natural number)　
Find (a, b, c, d, n)

Q6-(2)
If x^12 + x^10 + 2x^8 + 2x^6 + x^4 - 8x^2 - 9 = 0,
what's x^10 + 2x^6 - 4x^5 + x^2 - 4x -77 ?
(x: complex number)