b一种sic Algebra and Graphing for Electric Circuits

可以绘制用于电路分析的许多不同方程式。Take for instance Ohm’s Law for a 1 kΩ resistor:

按照欧姆定律绘制此图。然后，绘制另一个表示2kΩ电阻的电压/电流关系的图。

可以绘制用于电路分析的许多不同方程式。以欧姆法律为例，以获取与12伏来源相关的可变电阻器：

按照欧姆定律绘制此图。

Observe the following equivalence:

Since all operations are the same (multiplication) and reversible, the parentheses are not needed. Therefore, we may write the expression like this:

Of course, the simplest way to write this is 4⁵, since there are five 4’s multiplied together.

扩展这些表达式中的每一个，因此也没有指数：

3⁵×3²=

10⁴×10³=

8²×8³=

20¹×20²=

一种fter expanding each of these expressions, re-write each one in simplest form: one number to a power, just like the final form of the example given (4⁵）。From these examples, what pattern do you see with exponents of products. In other words, what is the general solution to the following expression?

Observe the following equivalence:

It should be readily apparent that we may cancel out two quantities from both top and bottom of the fraction, so in the end we are left with this:

re-writing this using exponents, we get 4¹。

扩展这些表达式中的每一个，因此也没有指数：

\((\frac{3^5}{3^2})\) =

\((\frac{10^6}{10^4})\) =

\（（（\ frac {8^7} {8^3}）\）=

\（（\ frac {20^5} {20^4}）\）=

一种fter expanding each of these expressions, re-write each one in simplest form: one number to a power, just like the final form of the example given (4¹）。From these examples, what pattern do you see with exponents of products. In other words, what is the general solution to the following expression?

Observe the following equivalence:

It should be readily apparent that we may cancel out two quantities from both top and bottom of the fraction, so in the end we are left with this:

佛llowing the rule of \((\frac{a^m}{a^n} = a^{m-n})\), the reduction of \((\frac{4^2}{4^3})\) should be 4^-1。许多学生发现这令人困惑，因为指数的直观概念（本身要乘以多少次）在这里失败。在世界上，我们如何将4乘以-1次？！

扩展这些表达式中的每一个，因此也没有指数：

\((\frac{3^2}{3^5})\) =

\（（（\ frac {10^4} {10^6}）\）=

\（（（\ frac {8^3} {8^7}）\）=

\((\frac{20^4}{20^5})\) =

一种fter expanding each of these expressions, re-write each one in simplest form: one number to a power, just like the final form of the example given (4^-1）, following the rule \((\frac{a^m}{a^n} = a^{m-n})\)。从这些示例中，您能想到什么易于理解的定义来描述负面指数？

另外，扩展以下表达式，因此没有指数，然后按照规则\（（（\ frac {a^m} {a^n} = a^a^{m-n}）\）重写指数形式的结果。

What does this tell you about exponents of zero?

When evaluating (calculating) a mathematical expression, what order should you do the various expressions in? In other words, which comes first: multiplication, division, addition, subtraction, powers, roots, parentheses, etc.; and then what comes after that, and after that?

佛llow proper order of operations to evaluate these expressions:

$$ \ frac {13 + 2} {3} + 8 = $$

$$ 25 +（3 + 2）^2 x 2 = $$

佛llow proper order of operations to evaluate these expressions:

$$\frac{15 - 3}{3} + 7 = $$

$$20 + (1+3)^2 x 3 = $$

在评估这样的表达式时，遵循适当的操作顺序非常重要。否则，正确的结果将无法得出：

To show what the proper order of operations is for this expression, I show it being evaluatedstep by stephere

对以下每个表达式执行相同的操作：

10− 25 × 2 + 5

− 8 + 10³× 51

12⁴×（3 + 11）

21^（7-4）×40

\(log \sqrt{6 + 35^2}\)

\(\sqrt{(\frac{220}{16}-2.75)} x 2\)

佛otnotes:

by the way, this is a highly recommended practice for those struggling with mathematical principles:记录每个步骤通过重写表达方式。尽管它需要更多的纸和更多的精力，但它会使您免于不必要的错误和挫败感
呢

Perform the following calculations:

Perform the following calculations:

The equation for calculating total resistance in a parallel circuit (for any number of parallel resistances) is sometimes written like this:

re-write this equation in such a way that it no longer contains any exponents.

一种functionis a mathematical relationship with an input (usually x) and an output (usually y). Here is an example of a simple function:

显示任何给定功能的模式的一种方法是使用数字表。完成此表的给定值x的值：

X 2X1 0 1 2 3 4 5

显示任何给定功能的模式的一种更常见的（和直觉）的方法是图形。完成此图的相同函数y = 2x 1.考虑轴上的每个划分为1个单位：

理解指数的著名说明性故事是这样的：

一种pauper saves the life of a king. In return, the king offers the pauper anything he desires as a reward. The pauper, being a shrewd man, tells the king he does not want much, only a grain of rice today, then double that (two grains of rice) the next day, then double that (four grains of rice) the next day, and so on. The king asks how long he is to give the pauper rice, and the pauper responds by saying one day for every square on a chess board (64 days). This does not sound like much to the king, who never took a math course, and so he agrees.

不过，在短时间内，国王发现自己破产了贫民，因为大米的数量非常大。这就是指数函数的性质：它们在x中的增长幅度非常大。

Graph the pauper’s rice function (y = 2^X）, with each division on the horizontal axis representing 1 unit and each division on the vertical axis representing 100 units.

Match each written function (y = …) with the sketched graph it fits best:

Match each written function (y = …) with the sketched graph it fits best: