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More negative exponent intuition

Video transcript

I don't think we can do enough videos on why raising something to a negative exponent is equivalent to 1 over that base raised to the positive exponent, I should say. And to get more intuition about why this makes sense, I look at different powers of 2, and then think about what makes sense as we go to exponents below 0, integer exponents below 0. So let's start with 2 to the third power. Well, 2 to the third power is 2 times 2 times 2, which of course is equal to 8. Now what about 2 to the second power? Well that's going to be 2 times 2, which is of course equal to 4. And to go from 2 to the third to 2 to the second power, what happened here? Well, we divided by 2. Now, let's keep going. What about 2 to the first power? Well, that's just 2, and once again, to go from 2 squared to 2 to the first power, we divided by 2. Now things are going to get interesting. 2 to the 0-th power, and actually this will help to build the intuition of why it's something nonzero to the 0-th power is defined to be 1. Well, so far, every time we decremented the exponent by 1, we essentially divided by 2, so we should divide by 2 again. So if we divide by 2 again, we get 1. And this is part of the motivation of why 2 to the 0 power should be equal to 1. But let's keep going. What should 2 to the negative 1 power be, if we want to be consistent about continuing to divide by 2? Well, we divide by 2 again, and so this is going to be equal to 1/2. Notice, 2 to the negative 1 is 1/2. 2 to the 1 is equal to 1. This is equal to the reciprocal of this. Let's keep going. This is fun. So what should 2 to the negative 2 power be? Well, we should divide by 2 again. Divide by 2 again, you get to 1/4. I think you see the pattern. 2 to the negative 3 power, well we should divide by 2 again. And we get to 1/8, which is the reciprocal of 2 to the third power. So once again, another way to think about why negative exponents are about taking reciprocals. Taking something to the negative exponent is equivalent to 1 over taking that same base to the positive exponent.