Why every number power 0=1 ?
The rule that any non-zero number raised to the power of 0 equals 1 can be understood through several mathematical principles. Here are a few explanations:
Definition of Exponents: The exponentiation of a number can be defined in terms of division. For any non-zero number ( a ): [ a^n = a^{n-1} \cdot a ] If we let ( n = 1 ), we have: [ a^1 = a^{0} \cdot a ] Dividing both sides by ( a ) (assuming ( a \neq 0 )): [ a^0 = \frac{a^1}{a} = \frac{a}{a} = 1 ]
Pattern in Powers: If you look at the pattern of powers of a number, you can see that as you decrease the exponent by 1, you are dividing by the base: [ a^3 = a \cdot a \cdot a ] [ a^2 = a^3 / a ] [ a^1 = a^2 / a ] [ a^0 = a^1 / a = 1 ] This pattern holds for any positive integer exponent.
Limit Approach: You can also think about the limit of ( a^n ) as ( n ) approaches 0. For any non-zero number ( a ), as ( n ) approaches 0, ( a^n ) approaches 1.
Algebraic Consistency: Defining ( a^0 = 1 ) ensures that the laws of exponents remain consistent. For example, the law ( a^m \cdot a^n = a^{m+n} ) holds true for all integers ( m ) and ( n ) if we define ( a^0 = 1 ).
Special Case of Zero: It's important to note that ( 0^0 ) is a special case and is often considered indeterminate in mathematics, though in some contexts it is defined as 1 for convenience.
In summary, the definition of exponents, the patterns observed in their behavior, and the need for consistency in mathematical operations all support the conclusion that any non-zero number raised to the power of 0 equals