This can be "handy" if you need some result to work in all cases such as the binomial theorem. See also What is 0 to the 0 power? How is that proved? What is the difference between power and the exponent?
Varthan The exponent is the little elevated number. In this case, 3 is the exponent, and 2 3 the entire expression is a power. Math Lessons menu. Hint: it has to do with a "recipe" that many math lessons follow. The do's and don'ts of teaching problem solving in math Advice on how you can teach problem solving in elementary, middle, and high school math.
How to set up algebraic equations to match word problems Students often have problems setting up an equation for a word problem in algebra. This article explains some of those relationships. And we've seen this before. You go to 2 to the third, you start with the 1, and then multiply it by 2 three times. So times 2 times 2 times 2. This is going to give us positive 8.
And you probably see a pattern here. Every time we multiply by or every time, I should say, we raise 2 to one more power, we are multiplying by 2. Notice this, to go from 2 to the 0 to 2 to the 1, we multiplied by 2. I'll use a little x for the multiplication symbol now, a little cross. And then to go from 2 to the first power to 2 to the second power, we multiply by 2 and multiply by 2 again.
And that makes complete sense because this is literally telling us how many times are we going to take this number and-- how many times are we going take 1 and multiply it by this number? And so when you go from 2 to the second power to 2 to the third, you're multiplying by 2 one more time. And this is another intuition of why something to the 0 power is equal to 1.
If you were to go backwards, if, say, we didn't know what 2 to the 0 power is and we were just trying to figure out what would make sense, well, when we go from 2 to the third power to 2 to the second, we'd be dividing by 2. We're going from 9 to 4. Then we'd divide by 2 again to go from 2 to the second to 2 to the first. And then it seems like we should just divide by 2 again from going from 2 to the first to 2 to the 0. Add a comment. From there, you can segue into negative exponents, if you'd like.
Just keep dividing by 2. But I was an odd kid who enjoyed formal logic and whose idea of fun was teaching my little sister simultaneous equations. She's now doing a maths PhD. Raphael Raphael 2, 21 21 silver badges 34 34 bronze badges. This is also the only way to wrap your head around negative powers. Incidentally, the best way to teach fractions is by having pizza for dinner. Rasmus MrGlass MrGlass 1 1 silver badge 4 4 bronze badges. When this occurred to me as an adult, I couldn't help but wonder if students who frequently ate pizza specifically, the round pie, cut into 8 sections thru the center developed better number sense.
The best we can do is to convince a child of the following facts: 1. Some of the other answers provide good ways to convince a child of these facts. Compare this to the following. BusyAnt 2 2 silver badges 14 14 bronze badges. Tsuyoshi Ito Tsuyoshi Ito 1, 11 11 silver badges 18 18 bronze badges.
Just write it out for some small values of a,b. I didn't come up with "convince a kid"; that's in direct response to " I do not think that it is possible to convince a child ". This can all be done using positive integers, it doesn't even need fractions or negative numbers. It's undefined by default. Show 2 more comments. Ross Millikan Ross Millikan k 26 26 gold badges silver badges bronze badges.
Asaf Asaf 4 4 bronze badges. Said identity should be true for any base. It isn't intuitive to their world view, and abstractions are at times challenging for people, especially kids. Anyway, my daughter is now 14 and I think gets it. I up-voted your answer, it is a valid approach, and more clearly stated than the comment, although I might think about reordering it, depending which part of those steps they intuitively find agreement with.
Matt Calhoun Matt Calhoun 4, 27 27 silver badges 51 51 bronze badges. Winther Sunny Sunny 1 1 1 bronze badge. Bill R Bill R 1. You can say that combining to no number i. Now, what's the multiplicative identity? Well, it's the only number which can be multiplied by any other number without changing that other number. So, the reason that any number to the zero power is one is because any number to the zero power is just the product of no numbers at all, which is the multiplicative identity, 1.
It's exciting to me that you asked this question. The fact is these rules are presented as somewhat arbitrary, but there is always, always well almost always good reason for them. Keep it up! If it ever sounds arbitrary then hound your teacher. If your teacher can't give you compelling reasons why something is true, hound us or hound Google. Okay, enough, onto your question:. Mathematics was initially developed to describe relationships between everyday quantities generally whole numbers so the best way to think about powers like a b 'a' raised to the 'b' power is that the answer represents the number of ways you can arrange sets of 'b' numbers from 1 to 'a'.
For example, 2 3 is 8. There are 8 ways to write sets of 3 numbers where each number can be either 1 or 2: 1,1,1 1,1,2 1,2,1 2,1,1 2,1,2 2,2,2 1,2,2 2,2,1. So what does 3 0 represent? It is the number of ways you can arrange the numbers 1,2, and 3 into lists containing none of them! How many ways are there to place a penny, a nickel, and a quarter on the table such that no coins are on the table?
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