why is a number raise to the power zero always given to be equal to one.?
Answers
Short answer: because powers “mean” repeated multiplication by a number (how many times given by the power).
So, to multiply something by a0 means to multiply by a zero times—that is, not to multiply at all, and thus the result is whatever number we started with:
x×a0=x
There’s only one number in a field with this property: the number one. So, a0=1 for any number a (even when a=0 ).
The arguments tend to start when we talk about 00 . Some would (wrong-mindedly) argue that zero to any power is zero, and thus 00 should be zero. But zero to a negative power is, in fact, undefined, so this argument has no basis.
A second problem is that once Calculus rolls around, students are told that 00 is an “indeterminate form,” which is wrong in several ways and is a side-effect of the limit pedagogy in calculus (akin to reading some English instructions that were translated from their original Japanese by way of Russian…which is why Calculus confuses so many students).
What they mean is something like (0+)0+ , which itself isn’t actually indeterminate, either. It’s divergent, though it can cause an indeterminate statement, which is the circuitous path by which it is called an “indeterminate form” in the first place: the misapplication of a term to an ill-expressed situation.
Step-by-step explanation:
- Well, it's the only number which can be multiplied by any other number without changing that other number. ,So, the reason that any number to the zero power is one is because any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1.