Section: 零和无能者的力量 | 9.3 0和负指数的力量-interactive

Section outline

Microscopic Numbers
::微小数字

Biologists and doctors regularly work with cells and other structures that are too small to be seen without the aid of a microscope. One of the ways the health of a cell is determined is by the size. Cells which are abnormally large can be an indicator of a serious illness. How do you think the size of a cell is measured?
::生物学家和医生经常与细胞和其他结构合作,这些细胞和其他结构太小,没有显微镜的帮助是看不到的。细胞健康的一个决定因素是大小。异常大的细胞可以作为严重疾病的一个指标。你认为如何测量细胞的大小?

When do you think you would use a microscopic ruler and when do you think you would use relative sizing?
::你认为什么时候会使用显微标尺? 什么时候会使用相对的缩略法?

The Zero Power
::零动力

Before you get to the microscopically small, you need to start at zero. If an exponent means to multiply the base that number of times how can something be multiplied zero times? Use the interactive below to explore this idea.
::在进入小微缩微镜片之前,您需要从零开始。如果一个指数意味着将基数乘以多少倍,那么某物如何乘以零倍?使用下面的交互作用来探索这个想法。

INTERACTIVE

The Zero Power

Drag the sliders to vary the number in the base and the exponent.
::拖动滑动符以修改基数和指数数。

Using the relationships between each term, try to figure out the value in the question mark before pressing the button to reveal the answer.
::使用每个词之间的关系, 尝试在按下按钮以显示答案之前, 找出问题标记中的值。

Progress

0 / 4

Look at the pattern 2, 4, 8, 16, 32...
::看模式2,4,8,8,16,32...

This pattern can also be written as $\begin{align*}{2}^{1}\end{align*}$ , $\begin{align*}{2}^{2}\end{align*}$ , $\begin{align*}{2}^{3}\end{align*}$ , $\begin{align*}{2}^{4}\end{align*}$ , $\begin{align*}{2}^{5}\end{align*}$ ...
::这个模式也可以写为 21,22,23,23,24,25...

As the exponents go up, what is happening? Choose the correct answer.
::当推手上升时,发生了什么?选择正确的答案。

What Are Negative Powers?
::什么是消极大国?

Continuing the pattern metaphor from the interactive in the previous section, look at this pattern: 2, 4, 8, 16, 32,... the number before 2 in this pattern would be represented by 2 ⁰ and equal to 1, but what number would come before that? Use the interactive below to explore the idea of negative exponents.
::继续前一节交互式比喻中的比喻模式,请看这个模式: 2, 4, 8, 16, 32... 这个模式中, 2 之前的数字将代表 20, 等于 1, 但前面的数字会是多少? 使用下面的交互模式来探索负指数的概念。

INTERACTIVE

Negative Powers

Drag the sliders to vary the number in the base and the exponent.
::拖动滑动符以修改基数和指数数。

Using the relationships between each term, try to figure out the values in the question marks before pressing the button to reveal the answer.
::使用每个词之间的关系, 尝试在按下按钮以显示答案之前, 找出问题标记中的值。

CK-12 PLIX Interactive: Patterns with Powers
::CK-12 PLIX 互动:权力模式

Use the interactive and answer questions to see how patterns in positive exponents relate to negative exponents.
::使用互动和回答问题来了解积极指数模式与消极指数的关系。

Simplifying Negative Powers
::简化消极大国

Since negative exponents represent division , a number to a negative power represents a fraction of 1. Biologists and doctors often write the size of cells and other microscopic organisms using negative exponents. In general, the size of a human cell is about $\begin{aligned} 10^{- 6} \end{aligned}$ meters. This expression simplifies to the following:
::由于负指数代表分裂,从数到负能量代表了1的一小部分。生物学家和医生常常使用负指数写出细胞和其他微小生物体的大小。一般而言,人类细胞的大小约为10-6米。

\begin{aligned} 10^{- 6} = \frac{1}{10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10} = \frac{1}{10^{6}} = \frac{1}{1, 000, 000} meters \end{aligned}

::10 - 10=1106=11006=11 000 000米

Another way to think of this is to find the reciprocal of the fraction with the opposite value for the exponent. Another way to think of $\begin{aligned} 10^{- 6} \end{aligned}$ is as the reciprocal of 10 with an exponent of 6, which would be $\begin{aligned} \frac{1}{10^{6}} \end{aligned}$ . Use the interactive below to explore this idea.
::另一种思考方式是找到分数的对等值与指数的对等值。另一种思考方式是10-6的对等值与6的对等值,即1106。使用下面的互动来探索这个想法。

INTERACTIVE

Simplifying Negative Exponents

Drag the red point to observe simplifying a negative exponent.
::拖曳红色点以观察是否简化负引号。

Drag the blue slider to vary the base.
::拖动蓝色滑块以更改底部。

I n general, you never want to write a negative exponent. All exponents should be written as positive. To convert a negative exponent to a positive exponent, move the base to the other side of the fraction and write the opposite value of the exponent.
::一般来说,您从不想写负指数。所有指数都应该写为正。要将负指数转换为正指数,请将底指数移到分数的另一面,并写出指数的相反值。

Example
::示例示例示例示例

Simplify $\begin{aligned} {(2 x y)}^{- 3} \end{aligned}$
::简化 (2xy)- 3

The base in this example is $\begin{aligned} 2 x y \end{aligned}$ which can be written as the fraction $\begin{aligned} \frac{2 x y}{1} \end{aligned}$ . You can move the base to the denominator and write the opposite value of the exponent.
::此示例中的基数是 2xy, 可以写成分数 2xy1 。您可以将基数移动到分母, 并写入引号的相反值。

\begin{aligned} \frac{1}{(2 x y)^{3}} \end{aligned}

::1(2xy)3

This can be further simplified by distributing the exponent to the term using the Power of a Product Property .
::通过使用产品产权的权力分配该术语的引号,可以进一步简化这一术语。

\begin{aligned} \frac{1}{(2 x y)^{3}} = \frac{1}{2^{3} x^{3} y^{3}} = \frac{1}{8 x^{3} y^{3}} \end{aligned}

::1 2xy)3=123x3y3=18x3y3

CK-12 PLIX Interactive: Negative Exponents
::CK-12 PLIX 互动:负指数

Use the interactive and answer questions to explore negative exponents.
::使用互动和回答问题来探索负面提示。

Summary

Anything to the power of zero will equal 1. For example: $\begin{aligned} x^{0} = 1 \end{aligned}$
::零功率的任何东西等于 1。例如: x0=1

When a variable is raised to a negative exponent it is equal to 1 over that variable with a positive exponent. For example: $\begin{aligned} x^{- 2} = \frac{1}{x^{2}} \end{aligned}$
::当将变量加为负引号时,该变量的值等于该变量的1,并带有正引号。例如: x- 2=1x2

Section outline

Microscopic Numbers ::微小数字

The Zero Power ::零动力

What Are Negative Powers? ::什么是消极大国?

Simplifying Negative Powers ::简化消极大国

Microscopic Numbers
::微小数字

The Zero Power
::零动力

What Are Negative Powers?
::什么是消极大国?

Simplifying Negative Powers
::简化消极大国