sologuitar Posted December 1, 2022 Share Posted December 1, 2022 Let me begin by saying that I am doing a self-study of college algebra. I love mathematics. I regret not majoring in math back in my student days. With that said, I hope this forum will allow me to post questions that I get stuck with in my review of college algebra. Let a = any integer. Why does a^(0) = 1? Link to comment Share on other sites More sharing options...

Eise Posted December 1, 2022 Share Posted December 1, 2022 One simple way to see it: a^3 = (a^4)/a a^2 = (a^3)/a a^1 = (a^2)/a = a a^0 = (a^1)/a = a/a = 1 Does that make sense? 1 Link to comment Share on other sites More sharing options...

sologuitar Posted December 2, 2022 Author Share Posted December 2, 2022 14 hours ago, Eise said: One simple way to see it: a^3 = (a^4)/a a^2 = (a^3)/a a^1 = (a^2)/a = a a^0 = (a^1)/a = a/a = 1 Does that make sense? Let me see. You are saying that, for example, a^0 is the same as (a^1)/a = a/a = 1. How do you go from (a^1)/a to a/a? Here is my logic considering the fact that the numerator and denominator have the same base a. I understand that any variable has a power of 1. Yes? For example: x = x^1, y = y^1, z = z^1, etc. Back to my example. (a^1)/a = a^(1 - 1) = a^0 = 1. The jump from a^0 = 1 is not clear for me. Link to comment Share on other sites More sharing options...

swansont Posted December 2, 2022 Share Posted December 2, 2022 2 minutes ago, sologuitar said: How do you go from (a^1)/a to a/a? a^1 = a Link to comment Share on other sites More sharing options...

Sensei Posted December 2, 2022 Share Posted December 2, 2022 a^1, a^2, a^3 are easy - anybody can do such calc even in memory. But mathematicians want to have universal functions, in this case pow(a,x). So, take piece of paper and make graph of f(x)=a^x, with a couple well-known x (a is constant and can be used any integer >= 2). Then use x=0.5, 0.25, 0.2, 0.125, 0.1, 0.01 etc. and you should see curve goes closer and closer to 1. Then draw line between them. With a=2, and default x in range -2 to +2, you will get such curve: https://www.wolframalpha.com/input?i=f(x)%3D2^x If you want to change range, use f.e. https://www.wolframalpha.com/input?i=f(x)%3D2^x%2C+x+range+-10+to+10 (Wolfram Alpha, and Excel/Spreadsheet are must-have for mathematicians these days) (you can draw such graph in Excel/Spreadsheet too) Link to comment Share on other sites More sharing options...

sologuitar Posted December 2, 2022 Author Share Posted December 2, 2022 5 hours ago, swansont said: a^1 = a Ok. I get it now. 4 hours ago, Sensei said: a^1, a^2, a^3 are easy - anybody can do such calc even in memory. But mathematicians want to have universal functions, in this case pow(a,x). So, take piece of paper and make graph of f(x)=a^x, with a couple well-known x (a is constant and can be used any integer >= 2). Then use x=0.5, 0.25, 0.2, 0.125, 0.1, 0.01 etc. and you should see curve goes closer and closer to 1. Then draw line between them. With a=2, and default x in range -2 to +2, you will get such curve: https://www.wolframalpha.com/input?i=f(x)%3D2^x If you want to change range, use f.e. https://www.wolframalpha.com/input?i=f(x)%3D2^x%2C+x+range+-10+to+10 (Wolfram Alpha, and Excel/Spreadsheet are must-have for mathematicians these days) (you can draw such graph in Excel/Spreadsheet too) Thanks but I am not trying to get technical here. I am simply reviewing material learned long ago. In fact, I think there are video clips on YouTube that explain this concept without using too much mathematical jargon. Link to comment Share on other sites More sharing options...

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