One of the great things about mathematics is that its rules build upon one another, using simple mathematical operations to prove more complex mathematical truths. Raising a number to the power of zero is no exception — you can prove that n0 = 1 by relying on simpler mathematical properties that you already know.
In this case, the two properties you need to know are
- nx × ny = nx+y
- The associative property of multiplication: (xy)z = x(yz)
Equation (a) is easy enough to show simply by choosing a couple of exponents and writing out the entire equation without using exponents, like this:
n3 × n4 = (n × n × n) × (n × n × n × n)
Because of the associative property of multiplication [see (b) above], you know that you can eliminate the parentheses and arrive at this:
n3 × n4 = n × n × n × n × n × n × n = n7
No matter what numbers or what exponents you try (unless you use zero as the base number), nx × ny = nx+y always.
With these two simple properties, you can better understand how raising to the power of zero works. Using what you've learned above, solve this equation:
n4 × n0 = ???
Because of (a) above, you know that
n4 × n0 = n4+0 = n4
The only way that n4 × n0 = n4 is if n0 = 1. Plugging real, non-zero numbers into an equation like this will yield the same results.
If you understand how negative exponents work, you could also take a different route to prove that n0 = 1. (Hint: n–x = 1/nx) Pick any non-zero number for n and solve this equation:
n–5 × n5 = ???
I'll leave it to you to figure it out.
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