Solving A Homemade Exponential Equation

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[@vladimirkaplun5774]

I do not see what you see. if c1 there is only one

[@briancatanzaro6053]

I am happy with keeping it "real".

[@seanfraser3125]

Raise both sides to the 5th power:
x^(15x^15) = 2^(5*32) = 2^(5*2^5)
Using exponent rules, we can rewrite this equation like this:
(x^15)^(x^15) = (2^5)^(2^5)
The function f(u) = u^u is one-to-one for u > 1, so the only solution to u^u = 32^32 is the obvious one, u=32. Thus
x^15 = 2^5.
Taking the 15th root of both sides gives us x = 2^(1/3).

[@walterwen2975]

Solving A Homemade Exponential Equation: x^(3x¹⁵) = 2³²; x =?
[x^(3x¹⁵)]⁵ = (2³²)⁵ = (2^2⁵)⁵, x^(15x¹⁵) = 2^[5(2⁵)]
2^[5(2⁵)] = 2^[15(2¹⁵⸍³)/3] = (2¹⸍³)^[15(2¹⸍³)¹⁵] = (³√2)^[15(³√2)¹⁵]
x^(15x¹⁵) = (³√2)^[15(³√2)¹⁵]; x = ³√2
Answer check:
x = ³√2: x^(3x¹⁵) = (³√2)^[3(³√2)¹⁵] = [(³√2)³]^[(³√2)³]⁵ = 2^2⁵ = 2³²; Confirmed
Final answer:
x = ³√2

[@Psykolord1989]

Before watching:
Looking at the problem before actually going into calculations, my first thought was 2^(1/3) power, i.e. cube root of 2. Why? Because of the exponents on the left.
3(x^15) struck me as awfully specific, and 2^5 = 32. So, I started with this idea and went from there.
Plugging in X=2^(1/3) gives us x^15 = (2^(1/3))^15. Recalling that (a^m)^n = a^(mn), we can simplify to 2^(15/3) = 2^5 = 32
This is the exponent we want. We now have (2^(1/3))^(3*32) = 2^(32*3/3) = 2^32.
This is admittedly doing the problem quite literally backwards. This strategy will very likely not work for most people; it just happened to work for me this time. (And this is of course not counting any complex solutions).
Still, at the end of the day, we arrive at x = 2^(1/3)

[@vighnesh153]

x^x = y^y => x = y is one of the solutions. There can be other solutions as well. For example. x = 1/2 and y = 1/4 are also possible solutions where x != y. How to find other solutions?

[@rakenzarnsworld2]

x = 2^(1/3)

[@scottleung9587]

Got it!

[@giuseppemalaguti435]

x=e^(W(160ln2)/15)=2^(1/3)

[@yoav613]

Nice! Time is runing and time is money😂💯

[@SyberMath]

Sure is!

[@phill3986]

👍🔥😁✌️👏👏✌️😁🔥👍

[@SidneiMV]

x³^x¹⁵ = 2^2⁵
x³ = 2 => *x = ∛2*
but if you don't believe it..
x³^x¹⁵ = 2^2⁵
x¹⁵lnx³ = 2⁵ln2
x¹⁵lnx¹⁵ = 2⁵ln2⁵
W(x¹⁵lnx¹⁵) = W(2⁵ln2⁵)
lnx¹⁵ = ln2⁵
x¹⁵ = 2⁵ => x = ∛2

[@AshuTosh-mx1gr]

145th viewer 😅

[@SyberMath]

Wow! 😁