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.
Explain with words and an example how any number raised to the zero power is 1?
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.