Exponent rules and Euler's identity question.

[D_Winds]

Hey,

I've got a quick math question, but I'm not sure if the logic is valid. Does:

a^(b*c) = d^(e*c) ?

Testing it with numbers, it seemed to work, but I'd like more knowledgeable confirmation


Now for Euler's identity, something doesn't rationalize well with me:

e^(i)(pi) = -1

This was apparently derived from:

e^(i)(pi) = cos(pi) + isin(pi)

Where cos(pi) = -1 and sin(pi) = 0

But if that's the case, then doesn't the make i irrelevant? Couldn't any value be replaced with that to obtain the same value since it is being multiplied by 0? If:

e^(x)(pi) = -1 + x(0), then that implied to me at least that e^anything*pi = -1

 

[D_Winds]

There are some numbers a, b, d, and e for which your equation is true, but it is not generally valid as testing with numbers shows.

Your example doesn't quite follow my idea though. I'll elaborate upon my query. If:

a^b = d^e
Then would a^(b*c) = d^(e*c) ?

Your example shows 2^2*2 = 3^4*2, but 2^2 =/= 3^4.

Ex.
2^6 = 4^3
64 = 64
2^(6*5) = 4^(3*5)

i is on BOTH sides of the equation so it is not irrelevant. Nothing is being multiplied by 0 on the left side of the equation.

e^(i)(pi) = e^(i)(pi)

e^(i)(pi) = cos(pi) + (i)sin(pi)

e^(i)(pi) = -1 + 0 = -1

Since "i" (or any other variable) is brought down from being an exponent and multiplied by sin(pi), making the product 0, this variable appears irrelevant. So what if it is not on left side of the equation, if the equivalent expression includes such a multiplication action.

e^(0)(pi) = 1 =/= -1

Then we are left with either:

e^(x)(pi) = 1 if x = 0.
e^(x)(pi) = -1 if x is non-zero.

 

[mmm4444bot]

I'll elaborate upon my query. If: a^b = d^e

Then would a^(b*c) = d^(e*c) ?

Ahh, you neglected (in your original post) to state the condition that a^b = d^e.

Knowing this condition allows us to say that a^(b*c) equals d^(e*c) for any set of values that lead to defined expressions.



There is a property of exponents that tells us:

a^(b*c) = (a^b)^c

d^(e*c) = (d^e)^c



Applying this property to the equation gives us:

(a^b)^c = (d^e)^c



But a^b = d^e, so, after substituting one for the other, both powers above are the same:

(a^b)^c = (a^b)^c

 

[D_Winds]

a^b = d^e
Then would a^(b*c) = d^(e*c) ? That is not the question you asked originally.

Sorry about that. I thought it was evident with just one line, but I guess not.


I'm still having some trouble Jeff. So to put it simply, are you saying:

e^(x)(pi) =/= cos(pi) + (x)sin(pi) for all values of x?

So "x" cannot be multiplied by sin(pi) for simplification? This is not a general rule (like a+b+c = b+c+a)?

 

[Deleted member 4993]

Another example:

2 * 2 = 2 + 2

However we cannot conclude:

x * x = x + x

Cannot replace a specific number (2) with a general variable (x).

In your case,

Cannot replace a specific number (i) with a general variable (x).