An Important Equation Appeared In Olympiads | Solving a^9 + a^6 = 36 | Aman Sir

In today's video, we will be solving an equation a^9 + a^6 = 36 Find the value of a.

this question was asked in Maths Olympiad Exam.

This is a very important question because you will learn a lot from this Maths Olympiad question, and it will be an excellent question to boost your Maths Olympiad preparation.

How will you solve this Maths Olympiad Problem?

Give it a thought and apply concepts that you learned to solve this Maths Olympiad Problem.

If you are unable to solve it, let's check out how Aman sir will make this Maths Olympiad Problem very easy to solve.

Check out the complete video to know the solution to the Maths Olympiad Problem.

An Important Equation Appeared In Maths Olympiad Questions | Bhannat Maths | Aman Sir Maths

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[madraxninja]

I solved this in 2min but not by your method 😊
It's so easy :
a⁹+a⁶=36
a⁹+a⁶= 3³+3² ( 36 can be written as<--)
By comparing both equations
a⁹ = 3³. , ,, And a⁶= 3²
Or
a= 3³/⁹ = 3⅓
a = 3²/⁶ = 3⅓
Hence Solved😊😊

[programmingwithkartik]

Sabse easy solution deko bhai
A⁹+a⁶=36
So,
(A³)³ + (a³)² = 27 + 9
(A³)³ + (a³)² = 3³ + 3²
So on comparing
A³=3
Hence value of a = 3^1/3

[utuberaj]

Nice one. You may have used the SYNTHETIC DIVISION for dividing by the factor (t-3)- this saves time- and that's critical in an Olympiad.

[shrayanpramanik]

To those who'd like to feel the level of Math Olympiads,
This question is probably the easiest one could get in city level Olympiads in India.

[KrishnaVerma-yqqy]

a⁹+a⁶= 36
a⁶(a³+1)=36
a⁶(a³+1)= (2×3)²
a⁶(a³+1)=2² × 3²
On comparing both sides,
a=³√2 or a= ³√3
Put both values in eq....
And only (a= ³√3) will satisfy....

[preet__mehta]

I did it in 30 seconds. As I saw powers 9 and 6, immediately a thought came in my mind that I should take a^3=t. So it will be t^3 + t^2=36
t^2 (t+1)= (3)^2 (3+1)
So t=3,
a^3=3.
a=(3)^1/3

[shweta]

It is also solved by heat and trial method when we expressed 36 in the form of 3 square and 2 square and then compare

[amitabhanand]

CHAT GPT😐
Solve for a :- a^9 + a^6 = 36

This equation is not easily solvable by simple algebraic methods. However, we can use numerical methods to find the solution.

One common method is to use Newton-Raphson method, which involves starting with an initial guess for the solution and iteratively updating the guess using the formula:

x(n+1) = x(n) - f(x(n))/f'(x(n))

where x(n) is the nth guess, f(x(n)) is the value of the equation at that guess, and f'(x(n)) is the derivative of the equation at that guess.

For this equation, the function f(a) = a^9 + a^6 - 36 and its derivative f'(a) = 9a^8 + 6a^5.

We can start with an initial guess of a = 2 and iterate the formula until we get a desired level of accuracy.

Here is an example of four iterations:

n=0: x(0) = 2; f(x(0)) = 2^9 + 2^6 - 36 = 32; f'(x(0)) = 9 * 2^8 + 6 * 2^5 = 992; x(1) = x(0) - f(x(0))/f'(x(0)) = 2 - 32/992 = 1.96875

n=1: x(1) = 1.96875; f(x(1)) = 1.96875^9 + 1.96875^6 - 36 = 13.3301; f'(x(1)) = 9 * 1.96875^8 + 6 * 1.96875^5 = 667.47; x(2) = x(1) - f(x(1))/f'(x(1)) = 1.96875 - 13.3301/667.47 = 1.96739

n=2: x(2) = 1.96739; f(x(2)) = 1.96739^9 + 1.96739^6 - 36 = 0.00354; f'(x(2)) = 9 * 1.96739^8 + 6 * 1.96739^5 = 637.40; x(3) = x(2) - f(x(2))/f'(x(2)) = 1.96739 - 0.00354/637.40 = 1.96739

n=3: x(3) = 1.96739; f(x(3)) = 1.96739^9 + 1.96739^6 - 36 = f'(x(3)) = 9 * 1.96739^8 + 6 * 1.96739^5 = 637.40; x(4) = x(3) - f(x(3))/f'(x(3)) = 1.96739 - = 1.96739

Since the value of f(x(3)) is very close to 0, we can consider the solution to be x = 1.96739.

So, the solution for a is a = 1.96739.

[Riversarelife]

{3^(1/3)}is the solution. Consider a³ = t. and then simple cubic equation will be formed. By hit and trial t= 3 satisfies this equation. and from this, value of a will be obtained.
Edit 1 - I didn't know that sir used the same method, but i don't copy him, i did it myself with watching the solution by sir.

[motivationdskate]

Koi jo bhi bole but aman sir ke jesa math koi nahi samjha sakta. THE GOD

[kavyanshtyagi]

more videos like this sir appreciate it a lot

[devapriyaguharoy]

Very interesting presentation. Thanks !

[shashankjha]

What i did was just from reading the thumbnail i just did a quick method so what i did was i took a^3 common and let it be t after that we get t^2(t+1)= 36 and by hut and trial i got t = 3 so a^3 = 3 so a= 3cube root

[justbadcm]

Easier approach would be to divide 36 in factors and check one by one directly writing something like 27 + 9 wood work in this case would work in this case but there could be multiple answers for other equations

[ajal]

I also solved it by using a³ = t and got my answer correct

[lakhikantadas]

Nothing can be more wonderful than maths

[RiteshKumar-bpdm]

Sir when u get the value of t = 3 then u could put that in a^3=t, then u get a^3=3 and than cube root both side u would directly get 3^⅓

[vaibhavpatil_]

Sir I am currently in class 10 and I also solved this question very easily in 4-5 steps but at one step I used trial and error method that's why Ans came in 4-5 steps the answer is (3)^1/3

[arhamnadeem]

I solved it right after seeing thumbnail and opened video to verify my answer

[royalredbird]

I also solved this equation by your method. (I'm in 10th)