Ingenious Method | Solutions to Exponential Equation 4^x+64x-192=0 | Math Olympiad

This math tutorial video shows you how to solve the mixed exponential equation 4^x+64x-192=0 ingeniously. The tricks / techniques include exponent rules, substitution, and the monotonicity of the function y=x^x.

You are welcome to watch and subscribe to the free math tutoring channel: Math Tutoring by Dr. Liang:

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[piman]

A great example of how algebra can sometimes be beautiful!

[prabhudasmandal]

Thanks for the nice problem and its solution.

[comfortablesofa]

SSOOOO Beautiful! Thank you! Loved the challenge each step of the way

[dalek]

Good but you need to be careful when raising to certain powers as you did with -1/t. It may be that you need plus minus signs e.g if x^2=4=2^2 raising to the 1/2 power without plus minus only gives x=2 instead of x=2 or x=-2.

[icems.a.]

Tal como dice en su título, es un método muy ingenioso. Siga enseñándonos 👏👏

[BlipBlop]

Thank you for showing a^0=1 earlier. It is akin to showing 0!=1 and that 1 can't be a prime if unique factorization of integers is to be achieved (1^2=1^3=1^4 etc.).

[prabhudasmandal]

Later, if (x--3) is taken as t, then (2^2)^t=--t or 2^2=(--t)^1/t=(--t)^--1×--1/t=--(1/t)^--1/t.so, 2=--1/t or 2t=--1.or 2(x--3)=--1 or 2x =6--1=5 or x=5/2.ans.

[larslrs]

Obviously, a solution for x is between 2 and 3. This equation looks suspicious af in terms of powers of two. So the solution might be representable as binary. Lets try 2.5 Done