Question

how may be this so difficile to solve it
- i like,need asking this,bc. much my college was sure that x=4 is the right answer,... inclusive colleges engineer ,till i got the idea how can i proving that x=4 not is the right value of x
- there is the exponential equation

2^16
------- = 2^2^x
16^2

x = ?

Answer 1

all colleges was secure,sure that x=4 is the right value of x till i ve wrote this :

ok. i said we accept that 4 is right so than there are

2^16
-------- = 2^2^4
16^2

2^16
------------ = 2^2^4
(2^4)^2

2^16
--------- = 2^2^4
2^8

2^(16-8) = 2^2^4

2^8 = 2^2^4 make logarithm base 2 on both sides

log(2) 2^8 = log(2) 2^2^4 using this properties of logarithms log a^x = xlog a so we get

8log(2) 2 = (2^4)log(2) 2 => 8 = 2^4 so 8 = 16

- in this way with this proof was accepted that x=4 not is the right,correct value,answer and and hence was proven that the right correct answer is sure x = 3 bc. 8 = 2^3

q.e.d.

Answer 2

hi @mathmate wait your opinion about above wrote - and thank you

Answer 3

i think just a parentheses may be the key of this exercise so than on the left part may be wrote

----------- = (2^2)^x

how you think this @mathmate ?

Answer 4

\(\dfrac{2^{16}}{16^{2}} = \dfrac{2^{16}}{(2^4)^{2}} = \dfrac{2^{16}}{2^8} = 2^8\)

\(2^8 = 2^{2^x} \implies 2^x = 8 \implies x = 3\)

Answer 5

Well, the question boils down to answering the question
What is \(4^{3^2}\)
If you evaluated it as \(4^{3^2}=4^9=262144\), then it is correct because exponentiation is right associated.
So similarly, 4^3^2 should be evaluated similarly, but noting that 4^3^2 is a sloppy way of writing \(4^{3^2}\) even though they are mathematically the same.

Based on the above logic, x=3, as you had it. @jhonyy9 congrats.

Answer 6

@jhonyy9
2^2^2 should have been written 2^(2^2) for clarity.
(2^2)^2 is a different problem and has different results.
The rule for chained exponents is to evaluate from right to left, different from +,-,and *.

Answer 7

i think that my colleges ve tought it in tis way that there are (2^2)^x using the exponential properties that

(x^a)^b = x^(a*b) and in this way have got they (2^2)^4 = 2^(2*4) = 2^8

so i just supposed this for i can understanding how they ve got for x this value of 4 like a right answer

Answer 8

\((a^2)^3\neq a^{2^3}\)

Answer 9

I am with @IrishBoy123

Answer 10

@ganeshie8
@TheSmartOne
@Kainui

Answer 11

When you read:

a^b^c

it means:

a^(b^c)