现在我们将谈论权力和根源。为了讨论指数的想法， 让我们首先考虑乘法。乘法真的是一次做大量添加的方式。 让我们想一想。如果我要要求你一起加六个四个， 没有人在他们的右心中坐在那里，并加入4 + 4 + 4 + 4。

没有人会这样做。当然，你会做的只是乘以4x6。 重要的是要记住，在任何繁殖的行为中，真的是你做的一切都是大量的添加。 Much in the same way, exponents are a way of doing a whole lot of multiplication at once.

如果我问你把七十三的衣服ether, we wouldn't write 3x3x3. We wouldn't write out that long expression instead we would write three to the seventh. Fundamentally, three to the seventh means that we multiply seven factors of 三个乘以。所以，它是一个非常紧凑的表示法，可以表达大量乘法。

I hasten to add, the test will not expect you to compute that value. It's not gonna be a test question, calculate three to the seventh, 这不是在测试中。但是，您必须处理与其他数量相关的那些数量。 例如，使用指数规律来弄清楚三到第七，并且整个东西被平方，或将三到第五到第五个乘以， 或将它分开。

你必须使用它，但你不必计算它的价值。象征性地，我们可以说b到n意味着b的n因素 are multiplied together. So this is the fundamental definition of what an exponent is and right now I'll just say b is the base, n is the exponent and b to the n is the power.

现在这是现在的一个很好的定义，但正如我们所看到的那样，这个定义最终有点天真，我们要将它扩展为以后的模块。 为什么它天真？好吧，如果你想到它，B乘以多少因素， 这意味着n是计数号码。也就是说，它是一个正整数。

And so this definition this way of thinking about exponents is perfectly good as long as the exponents are positive integers. But as we will see in upcoming modules there are all kinds of exponents that are not positive integers. We'll talk about negative exponents and fraction exponents, all that. Let's not worry about that in this module.

在此模块中，我们只需坚持正整数。所以我们可以坚持这个指数是什么非常直观的定义。 首先，我们可以向数字或变量提供指数。我们已经看到了代数模块中具有权力的变量， especially in the videos on quadratics, where you have x squared. Notice that we can read that expression either as seven to the power of eight or 七到第八。

其中一个是完全正确的。请注意，我们有不同的方式谈论两三个指数。 Something to the power of two is squared and something to the power of three is cubed. 所以我们很少说，到三个力量的东西，我们永远不会对两者的力量说些什么。

That just sounds awkward. We would always say that thing squared. If one is the base, then the exponent doesn't matter. One to any power is one. 事实上，这种表达式，一个到N等于一个，它适用于所有N，这不限于正整数。

这实际上适用于数字线上的每个单个号码，所以如果您将其放入N，则在数字行上每一个号码，则一到N等于一个。 所以这是要记住的重要一点。如果零是基础，则零到任何正指数为零。 因此，对于N等于零，只要零是正的。事实上，这是真的，不仅是积极的整数， 正面部分也是如此。

It's true of everything to the right of zero on the number line. So don't worry about zero to the power of zero or zero to the power of negatives, you will not have to deal with this on the test. That gets into either illegal mathematics or other forms of mathematics that we don't need to worry about, so that's just gonna be something we can ignore. An idea we have already discussed in the integer properties in Algebra lessons, if an exponent is not written, we can assume that the exponent is one.

我们在Prime Iachionations中谈到了这一点，我们在代数模块中再次谈到了这一点。 Another way to say that is any base to the power of one means that we have only one factor of that base, so two to the one is two. 2 squared is 4, 2 cubed is 3 factors so that's 8. So again, we're using the exponent as a way to count the number of factors we have in the total product.

如果基础是负面的，会发生什么？如果我们开始向权力提高负数？ 好吧，负二对一个，当然会是负二个。负两个平方，那是负面的消极， 这将是正面的四个。如果我们乘以两个负面因素， 积极的时间消极给我们一个负八。

Multiply another factor of 2, we get negative 8 times negative 2 gives us positive 16. 多个因素2，我们得到负面32.并注意到我们在这里有一种交替模式。 我们从负面到积极的，负面，负面，负面消极。

So we get a negative to any even power is a positive number. And a negative to any odd power is negative. We'll talk more about this in the next video. This has implication for solving algebraic equations. 例如，等式x平方等于四具有两个解决方案，x等于2和x等于负二，因为那些平方中的任一个等于四。

By contrast, the equation x cubed equals eight has only one solution x equals positive two. 如果我们立方体正面阳性，我们得到阳性八，但如果我们立方体负面二，我们得到负八。 Notice also that an equation of the form something squared equals a negative has no solution.

因此，例如，x减去一个平方等于负四。好吧，我们无法展示任何东西并获得负四个。 所以这是一个没有解决方案的等式。但我们可以让立方体等于消极的东西。 That's perfectly fine. If something cubed equals negative one than that thing must equal negative one 然后我们可以解决x。

最后，就像了解你的时期表一样重要，所以要了解单位数字的一些基本权力。 So here's what I'm gonna recommend memorizing and knowing. And it's helpful actually to multiply these out step by steps to help you 记住他们。首先， I'll recommend knowing the powers of two up to at least two to the ninth.

And why all the way up to two to the ninth? Well, we'll be talking about this more when we talk about some of the rules for 指数。但是，实际上非常好，一段时间练习一次。 只需乘以两个并获得所有这些数字，就可以让您自己验证他们来自的地方。

Know the powers of three up to at least three to the fourth. The powers of four up to the fourth, the powers of five up to the fourth. Again, multiply all these out from time to time just to remind yourself of all these so that you really can remember them very well. 那么你应该知道，当然，从六到九到六到九个方块和立方体，为什么需要了解所有这些？

Well, again we'll talk about these more when we talk about some of the rules of exponents. And of course know all the powers of ten, and that was discussed in the multiples at ten lesson is very easy to figure out powers of ten here. Just adding zeros or for negative powers you're putting it behind the decimal point.

从根本上B到N，意味着B的N个因素乘以一起。这是指数的基本定义。 And it's very good as we move through the laws of exponents to keep in mind that fundamental definition of an exponent. One to any power is one. Zero to any positive power is zero.

偶数权力是积极的。奇怪的力量是奇数。 An equation with an expression with to an even power equal to a negative is no solution, but an odd power can equal a negative. 最后，了解单位数字的基本权力。