Why is a^0=1?

Why is a^0=1?

In mathematics, it is often encountered that several identical numbers are multiplied. For this reason, a power operation is introduced and expressed by the symbol a, such as 7^5 =7×7 ×7×7 × 7 ×7×7. The n in a is called the exponent, which refers to the number of times the same number is multiplied. According to the definition of power, what is a^0 (a≠0) equal to? If this expression is understood as the result of a multiplying itself by 0 times, what is this result? Not surprisingly, we cannot determine the answer because in the definition of power, the exponent is a positive integer. Therefore, when we generalize the exponent from a positive integer to 0, we cannot apply the original definition. But the curious thing is that an appropriate result for a^0 can be given with the help of the original definition of power. Let's go back to positive integer powers and see the operational properties of powers.

Start with a simple example: 7^5 ×7^3 =(7×7 × 7 × 7×7) × (7×7×7)=7^8. There is a simpler algorithm for this: 7^5 ×7^3 =7^(5+3)=7^8. Generally, we have a^m ·a^n =a^(m+n). Similarly, since 0 cannot be divisor, we also require a≠0. We see the way to calculate a^0 from the algorithm of power. In fact, taking the same exponent, we have: a^m ÷a^m =1, so we get a^0 =1 (a≠0).

Based on the above definition, we can also extend the exponent to negative integers. Because a^(-n) ·a^n =a^0=1, there is a^(-n)=1/a^n. Further, the exponent in the power can be extended to the fractional exponent. This only requires admitting that when the exponent is a fraction, the algorithm of power still holds. So there is a^(1/2), which means that the multiplication of two a^(1/2) equals a, that is, squared equals a, and yes, it is √a. Following the same line of thinking, we can give a definition of when the index is a general fraction. If you learn limits, you can also extend the exponent to any real number. However, fractional exponent and irrational exponent often have more restrictions on a.