Why Is Anything to the Power of 0 Equal to 1? (Zero Exponent Rule)

Step by step

1. 5^0: multiply by 5 zero times → 0?
   Five to the power zero. You multiply five by itself zero times. So is it zero?
2. 0? 1? 5?
   Zero? One? Five? Take a second.
3. 5^3 = 125 · 5^2 = 25 · 5^1 = 5
   Follow the pattern. Five cubed is one twenty-five. Five squared is twenty-five. Five to the first is five.
4. Each step: ÷ 5 → 5^0 = 5 ÷ 5 = 1
   Each step down, we divide by five. One more step: five divided by five is one. So five to the zero is one!
5. Keep going: 5^−1 = 1/5
   Keep going, and it still works: five to the negative one is one fifth.
6. 5^3 ÷ 5^3 = 1 (anything ÷ itself = 1)
   Here's the proof. Five cubed divided by five cubed is one. Anything divided by itself is one.
7. Subtract exponents: 5^(3−3) = 5^0
   But the rule for dividing powers says: subtract the exponents. Three minus three is zero. So it's also five to the zero.
8. Any nonzero number to the power 0 = 1
   Same thing, two answers. So five to the zero must be one. Any number except zero, to the power zero, is one.
9. 0^0: one pattern says 0, the other 1
   So what about zero to the power zero? Zero squared, zero cubed: all zero. But one, two, three to the zero: all one. The two patterns disagree, and mathematicians use different answers in different places.
10. Python: 0 ** 0 → 1 · try it yourself
    Ask a computer. In Python, zero to the power zero gives one. Try it yourself.
11. 100 × 1.05^0 = 100 (the start)
    Where you'll see it: growth formulas. One hundred times one point zero five to the zero is one hundred: the money at the start, before any growth.
12. On a test: 5^3/5^3 = 1 and = 5^0 → 5^0 = 1
    On a test: five cubed over five cubed is one. By the quotient of powers, it's also five to the three minus three, which is five to the zero. So five to the zero is one.