2³=8, 2²=4, and 2¹=2 are easy to explain with multiplication. But 2⁰ or 2⁻¹ can seem strange if you think of it as "multiplying by 2 zero times." In fact, the answer follows naturally from the pattern. Change the base with the slider and watch how the value changes as the exponent decreases by 1.
Pattern rule: Each time the exponent in aⁿ decreases by 1, the value is divided by a. From 2⁴=16 to 2³=8, 16÷2=8; from 2³ to 2², 8÷2=4. If this division rule continues, one more division after 2¹=2 gives 2÷2=1, so 2⁰=1. Divide once more and 1÷2=1/2, giving 2⁻¹=1/2.
In other words, a⁰=1 and a⁻ⁿ=1/aⁿ are not arbitrary rules; they arise naturally so that the laws of exponents continue to hold at zero and for negative exponents.