I am trying to learn maths and I don't understand the following: 1. Something multiplied by it's reciprocal equals 1. 2. Something divided by a number is the same as it being multiplied by its reciprocal. 3. A fraction multiplied by another fraction is the same as their numerators multiplied together divided by their denominators multiplied by each other. 4. A number multiplied by a number divided by a number multiplied by a number is the same as one of the numerators divided by the product of the denominators and the total of that multiplied by the other numerator. 5. A number to the power of a negative number is the same as it's reciprocal to the power of the same power with the opposite sign. 6. A number to the power of a fraction is the same as the root of that number to the degree of the denominator (not sure if I wrote that right) and the product of that to the power of the numerator. Can anyone explain why? Examples aren't helping me. I can see that examples and proofs work, I just don't understand why they work. For example I can see that 3/1 x 1/3 is the same as 1 because 3 x 1 = 3 and 3/3 = 1 but I don't understand why every fraction multiplied by its reciprocal equals 1. I'm mostly using Khan Academy. I've watched all the playlists in the Arithmetic and Pre-Algebra section.