节: 带有指数的代数表达式 | 1.3 带有指数的代数表达式

章节大纲

Exponents
::指数

Many formulas and equations in mathematics contain exponents. Exponents are used as a short-hand notation for repeated multiplication . For example:
::数学中的许多公式和方程式含有指数。指数被用作重复乘法的短手符号。例如 :

$\begin{aligned} 2 \cdot 2 & = 2^{2} \\ 2 \cdot 2 \cdot 2 & = 2^{3} \end{aligned}$

The exponent stands for how many times the number is used as a factor (multiplied). When dealing with integers , it's usually easiest to simplify the expression . For example, as shown here, we commonly just write 4 instead of $\begin{aligned} 2^{2}, \end{aligned}$ and 8 instead of $\begin{aligned} 2^{3} . \end{aligned}$
::引号表示数字是作为系数( 倍数) 的多少倍。在处理整数时, 通常最容易简化表达式。例如, 如此处所示, 我们通常只写 4, 而不是 22, 8 而不是 23 。

$\begin{aligned} 2^{2} & = 4 \\ 2^{3} & = 8 \end{aligned}$

However, we generally keep the exponents when we work with variables, because it is much easier to write $\begin{aligned} x^{8} \end{aligned}$ than $\begin{aligned} x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x . \end{aligned}$
::然而,我们通常在与变量合作时保留指数,因为写x8比写xxxxxxxxxxxxxxxxxxxxxxxxxx要容易得多。

To evaluate expressions with exponents, substitute the values you are given for each variable and simplify. It is especially important when working with exponents to substitute using " data-term="Parentheses" role="term" tabindex="0"> parentheses in order to make sure that the simplification is done correctly.
::要用表率来评价表达式, 请替换您给每个变量提供的值, 并简化。在与表率一起使用括号时, 使用括号替换特别重要, 以确保简化正确进行。

Evaluating an Expression Containing Exponents
::评估含有指数的表达式

The area of a circle is given by the formula $\begin{aligned} A = π r^{2} . \end{aligned}$ Find the area of a circle with radius $\begin{aligned} r = 17 i n c h e s . \end{aligned}$
::公式Ar2给定一个圆区域,以半径 r=17 英寸查找圆区域。

Substitute values into the equation .
::等式中的替代值。

$\begin{aligned} A & = π r^{2} Substitute 17 for r . \\ = π (17)^{2} \\ = 3.14 (17) (17) \\ A & \approx 907.92 Rounded to 2 decimal places. \end{aligned}$

::r2 替代 17 r. (17)2= 3.14(17)(17)A 907.92四舍五入到小数点后两位。

The area of the circle is approximately 907.92 square inches.
::圆的面积约为907.92平方英寸。

Evaluating a Multi-Variable Expression Containing Exponents
::评估含有多变量指数的多功能表达式

$\begin{aligned} Find the value of \frac{x^{2} y^{3}}{x^{3} + y^{2}}, when x = 2 and y = - 4. \end{aligned}$
::当 x=2 和 y4 时, 查找 x2y3x3+y2 的值。

Substitute the given values for $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ .
::替换 x 和 y 的给定值。

$\begin{aligned} \frac{x^{2} y^{3}}{x^{3} + y^{2}} & = \frac{(2)^{2} (- 4)^{3}}{(2)^{3} + (- 4)^{2}} Substitute 2 for x and - 4 for y . \\ = \frac{4 (- 64)}{8 + 16} (2)^{2} = 4, (- 4)^{3} = - 64, (2)^{3} = 8, (- 4)^{2} = 16. \\ = \frac{- 256}{24} \\ = \frac{- 32}{3} \end{aligned}$

::x2y3x3+y2=(2)2(-4)3(2)3+(-4)2x和y.=4(-648+16(2)2=4,(-4)3x3+y2=2,(-4)3(2)3+(-4)3+(-4)2 y.=4(-648+16(2)2=4,(-4)3x3+y2=8,(-4)2=16.*25624=323

Real World Application
::真实世界应用

The height "> $\begin{aligned} (h) \end{aligned}$ of a ball in flight is given by the formula $\begin{aligned} h = - 32 t^{2} + 60 t + 20 \end{aligned}$ , where the height is given in feet and the time $\begin{aligned} (t) \end{aligned}$ is given in seconds. Find the height of the ball at time $\begin{aligned} t = 2 seconds. \end{aligned}$
::飞行中的球的高度由公式 h32t2+60t+20给定, 高度以足为单位, 时间(t) 以秒为单位。在时间 t=2 秒时查找球的高度。

$\begin{aligned} h & = - 32 t^{2} + 60 t + 20 \\ = - 32 (2)^{2} + 60 (2) + 20 Substitute 2 for t . \\ = - 32 (4) + 60 (2) + 20 \\ = 12 \end{aligned}$

::há32t2+60t+2032(2)+60(2)+20替代品2, t. @32(4)+60(2)+20=12

The height of the ball is 12 feet.
::球的高度是12英尺

Example
::示例示例示例示例

Example 1
::例1

Find the value of $\begin{aligned} \frac{a^{2} + b^{2}}{a^{2} - b^{2}} \end{aligned}$ , for $\begin{aligned} a = - 1 \end{aligned}$ and $\begin{aligned} 5 \end{aligned}$ .
::查找 a2+b2a2-b2 和 a%1 和 5 的 a2+b2a2-b2 值。

Substitute the values of $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ in the following.
::以 x 和 y 的值替换以下数值。

$\begin{aligned} \frac{a^{2} + b^{2}}{a^{2} - b^{2}} = \frac{(- 1)^{2} + (5)^{2}}{(- 1)^{2} - (5)^{2}} Substitute - 1 for a and 5 for b . \\ \frac{1 + 25}{1 - 25} = \frac{26}{24} = \frac{13}{12} Evaluate and simplify expressions. \end{aligned}$

::a2+b2a2-b2=(-1)2+(5)2(-2-1)-2-(5)2替代品-1和b.1+251-25=2624=1312

Review
::回顾

Evaluate 1-8 using $\begin{aligned} x = - 1, y = 2, z = - 3, \end{aligned}$ and $\begin{aligned} w = 4 \end{aligned}$ .
::使用 x1, y=2, z3 和 w=4 来评价 1-8 。

$\begin{aligned} 8 x^{3} \end{aligned}$
::8x3

$\begin{aligned} \frac{5 x^{2}}{6 z^{3}} \end{aligned}$
::5x26z3

$\begin{aligned} 3 z^{2} - 5 w^{2} \end{aligned}$
::3z2-5w2

$\begin{aligned} x^{2} - y^{2} \end{aligned}$
::x2 - y2

$\begin{aligned} \frac{z^{3} + w^{3}}{z^{3} - w^{3}} \end{aligned}$
::z3+w3z3-w3 z3 -w3

$\begin{aligned} 2 x^{3} - 3 x^{2} + 5 x - 4 \end{aligned}$
::2x3-3x2+5x-4

$\begin{aligned} 4 w^{3} + 3 w^{2} - w + 2 \end{aligned}$
::4w3+3w2-2-2+2

$\begin{aligned} 3 + \frac{1}{z^{2}} \end{aligned}$
::3+1z2 3+1z2

For 9-10, use the fact that the volume of a box without a lid is given by the formula $\begin{aligned} V = 4 x (10 - x)^{2} \end{aligned}$ , where $\begin{aligned} x \end{aligned}$ is a length in inches and $\begin{aligned} V \end{aligned}$ is the volume in cubic inches.
::9-10时,使用公式V=4x(10-x)2给出无盖子的盒体体积这一事实,x是一个长度英寸,V是立方英寸的体积。

What is the volume when $\begin{aligned} x = 2 \end{aligned}$ ?
::x=2 时的音量是多少 ?

What is the volume when $\begin{aligned} x = 3 \end{aligned}$ ?
::x=3 时的音量是多少 ?

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

章节大纲

Exponents ::指数

Evaluating an Expression Containing Exponents ::评估含有指数的表达式

Evaluating a Multi-Variable Expression Containing Exponents ::评估含有多变量指数的多功能表达式

Real World Application ::真实世界应用

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Exponents
::指数

Evaluating an Expression Containing Exponents
::评估含有指数的表达式

Evaluating a Multi-Variable Expression Containing Exponents
::评估含有多变量指数的多功能表达式

Real World Application
::真实世界应用

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)