@AstroGuy99 Hi Richard, thank you so much for your comment and positive feedback, as always! 😊 I am so happy you enjoyed the video and I hope you have an amazing day/evening/night! 😊
Do you want to learn the next step, how to solve exponent EQUATIONS? Check out: ua-cam.com/video/qbA71MNkKuk/v-deo.htmlsi=XtAVQleTcEFBSWyb (a popular video on my channel) and learn the idea of "fractional exponents" too, within 4 mins! 🥳 Let's get to 1000 subscribers! 🥳 Happy Holidays to you and your family!!! 🥳🎊🎉
Hi @renesperb thank you so much for your comment! 😊 Yes, that is the mathematical way of exhibiting a^{0} = 1 using exponent laws. I think the main question is: how do you motivate a^{-1} = 1/a to someone who just understands positive exponents (i.e., understands a^n = a x a ... x a (n times) for n a positive integer)? If you have that, then the exponent sum laws imply a^0 = 1 as you explain. Alternatively, without using negative exponents, we could argue a^{1} = a^{1 + 0} = a^{1}*a^{0}, which means that a^{0} = 1. As always, thanks so much for your comments and for sharing your thoughts! I hope you have an amazing day/evening/night! 😊
@@MathMasterywithAmitesh One has to define at some point what a^(-1) means.Then one can see that this definition makes sense ,using the already known rules of calculating with powers. The same applies to a^(1/2) and similar expressions.
@@renesperb Hi! 😊 Yes, I think once someone understands the definition of a^x for all rational numbers x, then one sees that it is very natural. I think the main difficulty is possibly that the idea of a^n = a x a x ... x a (n times) is natural to very early learners of math for n a positive integer, but a^{-1} and a^{0} are less clear at the beginning (and probably aren't useful until algebra/precalculus). I think young kids (and other early learners) take time to wrap their head around the idea of multiplying something with itself "0 times" or "-1 times". I agree with you that once we define them and get used to it, it makes a lot of sense though! I hope you have an amazing day/evening/night! 😊
that's weird, I've always been taught n^0 is equal to 1. Is there any benefit to saying it equals to 0? I'm really failing to see the point in saying that instead of 1 lol
Hi @justinferland6129 I understand what you mean! I think once you get beyond a certain level with exponents and exponent laws, it is very natural that n^0 = 1 for any non-zero number n. I think the question is more: if you saw exponents for the first time, e.g., you learnt 2^n = 2 x ... x 2 (n times), what would you think 2^0 is? At this stage of someone's math education, it's conceivable they would think 2^0 = 0 since you are multiplying 2 with itself 0 times, so you should get "nothing". I don't think there is any benefit to saying 2^0 = 0 (after all it is wrong 😅) but it's just a misconception that could be natural at lower levels. I definitely remember a few students in high school (in addition to my math teacher in 5th grade) thinking this. The more general math convention is that "an empty product is equal to 0" (where "empty product" is taking the product of numbers over an empty set) which can also be counterintuitive. I hope you have an amazing day/evening/night! 😊
Very clearly presented. Thanks Amitesh.
@AstroGuy99 Hi Richard, thank you so much for your comment and positive feedback, as always! 😊 I am so happy you enjoyed the video and I hope you have an amazing day/evening/night! 😊
Do you want to learn the next step, how to solve exponent EQUATIONS? Check out: ua-cam.com/video/qbA71MNkKuk/v-deo.htmlsi=XtAVQleTcEFBSWyb (a popular video on my channel) and learn the idea of "fractional exponents" too, within 4 mins! 🥳
Let's get to 1000 subscribers! 🥳 Happy Holidays to you and your family!!! 🥳🎊🎉
You give a good motivation for 2^0 . A different possibility for any a>0 : a^0 = a^(1-1) = a^1/a^1 =1.
Hi @renesperb thank you so much for your comment! 😊 Yes, that is the mathematical way of exhibiting a^{0} = 1 using exponent laws. I think the main question is: how do you motivate a^{-1} = 1/a to someone who just understands positive exponents (i.e., understands a^n = a x a ... x a (n times) for n a positive integer)? If you have that, then the exponent sum laws imply a^0 = 1 as you explain.
Alternatively, without using negative exponents, we could argue a^{1} = a^{1 + 0} = a^{1}*a^{0}, which means that a^{0} = 1.
As always, thanks so much for your comments and for sharing your thoughts! I hope you have an amazing day/evening/night! 😊
@@MathMasterywithAmitesh One has to define at some point what a^(-1) means.Then one can see that this definition makes sense ,using the already known rules of calculating with powers. The same applies to a^(1/2) and similar expressions.
@@renesperb Hi! 😊 Yes, I think once someone understands the definition of a^x for all rational numbers x, then one sees that it is very natural. I think the main difficulty is possibly that the idea of a^n = a x a x ... x a (n times) is natural to very early learners of math for n a positive integer, but a^{-1} and a^{0} are less clear at the beginning (and probably aren't useful until algebra/precalculus). I think young kids (and other early learners) take time to wrap their head around the idea of multiplying something with itself "0 times" or "-1 times".
I agree with you that once we define them and get used to it, it makes a lot of sense though! I hope you have an amazing day/evening/night! 😊
that's weird, I've always been taught n^0 is equal to 1. Is there any benefit to saying it equals to 0? I'm really failing to see the point in saying that instead of 1 lol
Hi @justinferland6129 I understand what you mean! I think once you get beyond a certain level with exponents and exponent laws, it is very natural that n^0 = 1 for any non-zero number n.
I think the question is more: if you saw exponents for the first time, e.g., you learnt 2^n = 2 x ... x 2 (n times), what would you think 2^0 is? At this stage of someone's math education, it's conceivable they would think 2^0 = 0 since you are multiplying 2 with itself 0 times, so you should get "nothing". I don't think there is any benefit to saying 2^0 = 0 (after all it is wrong 😅) but it's just a misconception that could be natural at lower levels. I definitely remember a few students in high school (in addition to my math teacher in 5th grade) thinking this.
The more general math convention is that "an empty product is equal to 0" (where "empty product" is taking the product of numbers over an empty set) which can also be counterintuitive.
I hope you have an amazing day/evening/night! 😊