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Nope, not true. 0^4 = 0^5.
Yea, but can you really call this exponential? If you do, then just about every single equation is exponential. Example:
Did you get that maths book as a Christmas present?
In a book I'm reading an exponential equation is defined as an equation containing terms of the form a^x (a to the power of x), where x is a real number, a is a real number, a > 0 and a is not equal to 1.