What is the rule for exponents, how you can tell 2 different problems will be equal?

Question

like, 2^4=4^2
but 3^2 is not equal to 2^3
Is there a rule for this?

Answer 1

This is not common.

In general, you're looking for x^y = y^x.

This is equivalent to x^(y/x) = y, which in turn is equivalent to x^(1/x)=y^(1/y).

x^(1/x) has a maximum at e^(1/e) = 1.44466.... The function is increasing before that point and decreasing after. As a result, for each number there is another on the opposite side with the same value. 2^(1/2) = 4^(1/4) <--> 2^4 = 4^2 is the only example of this in the integers.

Answer 2

In general, powers don't work that way. You won't see x^y = y^x all the time

Now, with 2^4, that can be written as (2^2)(2^2), which can then be written as (4)(4) = 4^2. That is more of a special case because of the relationship of 2 and 4.

Answer 3

before doing something the bases would desire to be an identical, a sq. root is comparable to asserting a a million/2 ability so substitute ?2 to 2^(a million/2): 2^(4x + a million) = 2^(a million/2) Now that the bases are an identical, drop the powers and placed them into an equation: 4x + a million = a million/2 4x = -a million/2 x = -a million/8 And to respond to your different question, only about all those questions are stated as exponential equations.

Answer 4

if u break down the problem...

2^4 = 4^2
2^4 = (2^2)^2
2^4 = 2^4 u see they are just the same base.

Answer 5

2^4 = 2x2x2x2
4^2 = 4x4 = 2x2x2x2

You should not expect this to be the rule.

Answer 6

As one of your respoders explained, it is very rare that a^b = b^a.
The fact that it works for 2 and 4 can basically be viewed as a coincidence.

Here are the basic rules you need to know about exponents:
(x^a) (x^b) = x^(a+b)
(x^a) / (x^b) = x^(a-b)
(x^a)^b = x^(ab)

Note, however, that x^(a^b) is NOT equal to x^(ab).

Answer 7

There isn't much of a rule to this. You just have to know how to work with powers.