课程： 1.5 带有分数栏的代数表达式

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带有分数栏的代数表达式
Expressions Involving Fraction Bars
::包含分数栏的表达式

Fraction bars count as grouping symbols for PEMDAS , so we evaluate them in the first step of solving an expression . All numerators and all denominators can be treated as if they have invisible " data-term="Parentheses" role="term" tabindex="0"> parentheses around them. When real parentheses are also present, remember that the innermost grouping symbols come first. If, for example, parentheses appear on a numerator, they would take precedence over the fraction bar . If the parentheses appear outside of the fraction, then the fraction bar takes precedence.
::PEMDAS 的分数条计为组合符号, 所以我们在解析表达式的第一步中对它们进行评估。所有的分子和所有分母都可以被当做它们周围有无形的括号。当真的括号也存在时, 请记住最核心的组合符号排在前面。例如, 如果括号出现在数字上, 它们将优先于分数栏。如果括号出现在分数栏之外, 则分数栏居优先。

Evaluating Expressions
::评价表达式

Use the to evaluate the following expression: $\begin{aligned} \frac{z + 3}{4} - 1 \end{aligned}$ when $\begin{aligned} z = 2 \end{aligned}$
::用于评价以下表达式:z+34-1,当z=2时;z=2

We substitute the value for $\begin{aligned} z \end{aligned}$ into the expression.
::我们在表达式中替换 z 值。

$\begin{aligned} \frac{2 + 3}{4} - 1 \end{aligned}$

Although this expression has no parentheses, the fraction bar is also a grouping symbol, it has the same effect as a set of parentheses. We can write in the “invisible parentheses” for clarity:
::虽然这个表达式没有括号, 分数栏也是一个分组符号, 但它与一组括号具有相同的效力。我们可以在“ 隐形括号” 中写明:

$\begin{aligned} \frac{(2 + 3)}{4} - 1 \end{aligned}$

Using PEMDAS , we first evaluate the numerator, 2 + 3 = 5. Now we have:
::使用PEMDAS,我们首先评估分子,2+3=5。现在我们有:

$\begin{aligned} \frac{5}{4} - 1 \end{aligned}$

We can convert $\begin{aligned} \frac{5}{4} \end{aligned}$ to a mixed number:
::我们可以将54转换为混合数字:

$\begin{aligned} \frac{5}{4} = 1 \frac{1}{4} \end{aligned}$

Then evaluate the expression:
::然后评价这个表达方式:

$\begin{aligned} \frac{5}{4} - 1 = 1 \frac{1}{4} - 1 = \frac{1}{4} \end{aligned}$

Evaluating Multi-Variable Expressions
::评价多变量表达式

Use the order of operations to evaluate the expression $\begin{aligned} (\frac{a + 2}{b + 4} - 1) + b \end{aligned}$ when $\begin{aligned} a = 3 \end{aligned}$ and $\begin{aligned} b = 1. \end{aligned}$
::使用操作顺序来评价表达式 (a+2b+4-1) +b, 当 a=3 和 b=1 时。

Substitute the given values for $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ into the expression:
::将 a 和 b 的给定值替换为表达式 :

$\begin{aligned} (\frac{3 + 2}{1 + 4} - 1) + 1 \end{aligned}$

This expression effectively has nested parentheses (remember the effect of the fraction bar). The innermost grouping symbol is provided by the fraction bar. Evaluate the numerator $\begin{aligned} (3 + 2) = 5 \end{aligned}$ and denominator $\begin{aligned} (1 + 4) = 5 \end{aligned}$ first:
::此表达式实际上已嵌套括号( 记住分数栏的效果 ) 。最核心的分组符号由分数栏提供。先评价分子 (3+2)=5 和分母 (1+4)=5 :

$\begin{aligned} (\frac{3 + 2}{1 + 4} + 1) + 1 \\ = (\frac{5}{5} - 1) + 1 Now evaluate inside the parentheses. First, divide. \\ = (1 - 1) + 1 Next, subtract. \\ = 0 + 1 = 1 \end{aligned}$

:3+21+4+1+1)+1=(55-1)+1,在括号内进行评价。首先,除法=(1-1)+1Next,减去。=0+1=1=1

Using a Graphing Calculator
::使用图形计算计算器

Use a graphing calculator to evaluate the expression $\begin{aligned} \frac{3 x^{2} - 4 y^{2} + x^{4}}{(x + y)^{\frac{1}{2}}} \end{aligned}$ for $\begin{aligned} x = 2, y = - 1. \end{aligned}$

::使用图形计算器来评价表达式 3x2 - 4y2+x4( x+y) 12 x=2, y1 的表达式。

Store the values of $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ : 2 [STO] $\begin{aligned} x \end{aligned}$ , -1 [STO] $\begin{aligned} y \end{aligned}$ . (The letters $\begin{aligned} x \end{aligned}$ and $\begin{aligned} y \end{aligned}$ can be entered using [ALPHA] + [KEY] .) Input the expression in the calculator. When an expression includes a fraction, be sure to use parentheses: $\begin{aligned} \frac{(numerator)}{(denominator)} \end{aligned}$ .
::x 和 y: 2 [STO]x, -1 [STO]y. (字母 x 和 y 可以使用 [ALPHA] + [KEY] 输入输入。) 将表达式输入计算器。当表达式包含一个分数时, 请使用括号 : (计数器) (计数器) 。

Press [ENTER] to obtain the answer, $\begin{aligned} 24 \end{aligned}$ .
::新闻[主获得答案,24。

Example
::示例示例示例示例

Example 1
::例1

Use the order of operations to evaluate the expression $\begin{aligned} 2 \times (\frac{w + (x - 2 z)}{(y + 2)^{2}} - 1) \end{aligned}$ when $\begin{aligned} w = 11, x = 3, y = 1, \end{aligned}$ and $\begin{aligned} z = - 2. \end{aligned}$
::在 w=11, x=3, y=1, z=2时, 使用操作顺序来评价表达式 2x(w)+(x- 2z)(y+2)2-2- 1 。

Substitute the values for $\begin{aligned} w, x, y, \end{aligned}$ and $\begin{aligned} z \end{aligned}$ into the expression:
::将 w、 x、 y 和 z 的值替换为表达式 :

$\begin{aligned} 2 \times (\frac{11 + (3 - 2 (- 2))}{(1 + 2)^{2}} - 1) \end{aligned}$

This complicated expression has several layers of nested parentheses. One method for ensuring that we start with the innermost parentheses is to use more than one type of parentheses. Working from the inside, we leave the innermost grouping symbols as parentheses $\begin{aligned} () \end{aligned}$ . Next will be the “invisible brackets” from the fraction bar, write these as $\begin{aligned} [] \end{aligned}$ . The third level of nested parentheses will be the braces $\begin{aligned} {} \end{aligned}$ . Leave negative numbers in regular parentheses.
::此复杂表达式有几层嵌套括号。确保我们从最内部括号开始的方法之一是使用不止一种括号。从内部看,我们把最内部组别符号作为括号。下一步是分数栏的“ 隐形括号” , 将其写成 [ ]。嵌套括号的第三级将是括号 {}。在常规括号中保留负数。

$\begin{aligned} 2 \times {\frac{[11 + (3 - 2 (- 2))]}{[{(1 + 2)}^{2}]} - 1} Start by evaluating the parentheses. \\ (3 - 2 (- 2)) = 3 + 4 = 7 : (1 + 2) = 3 \\ = 2 {\frac{[11 + 7]}{[3^{2}]} - 1} Next, evaluate the square brackets. \\ = 2 {\frac{18}{9} - 1} Now evaluate the braces. Start with division. \\ = 2 {2 - 1} Finally, do the addition and subtraction. \\ = 2 {1} = 2 \end{aligned}$

::2[11+(3-2-2)][(1+2-2)-2-1}以评价括号开始。 (3-2)=3+4=7: (1+2)=3=2{[11+7][32]-1}下一步,评估方括号。=2{189-1}现在评估括号。以除法开始。=2{{2-1}最后,做加和减。=2{1}

Review
::回顾

For 1-3, use the order of operations to evaluate the expressions.
::1-3,使用操作顺序来评价表达式。

$\begin{aligned} \frac{2 \cdot (3 + (2 - 1))}{4 - (6 + 2)} - (3 - 5) \end{aligned}$

$\begin{aligned} \frac{4 + 7 (3)}{9 - 4} + \frac{12 - 3 \cdot 2}{2} \end{aligned}$

$\begin{aligned} \frac{(2^{2} + 5)^{2}}{5^{2} - 4^{2}} \div (2 + 1) \end{aligned}$

For 4-9, evaluate the expressions by substituting for the variables.
::4-9时,用变量替换来评价表达式。

$\begin{aligned} \frac{j k}{j + k} \end{aligned}$ when $\begin{aligned} j = 6 \end{aligned}$ and $\begin{aligned} k = 12 \end{aligned}$
::j=6 和 k=12 时jkj+k

$\begin{aligned} \frac{x + y^{2}}{y - x} \end{aligned}$ when $\begin{aligned} x = 2 \end{aligned}$ and $\begin{aligned} y = 3 \end{aligned}$
::x=2 和 y=3 时 x+y2y-x

$\begin{aligned} \frac{4 x}{9 x^{2} - 3 x + 1} \end{aligned}$ when $\begin{aligned} x = 2 \end{aligned}$
::x=2 时 4x9x2 - 3x+1

$\begin{aligned} \frac{z^{2}}{x + y} + \frac{x^{2}}{x - y} \end{aligned}$ when $\begin{aligned} x = 1, y = - 2, \end{aligned}$ and $\begin{aligned} z = 4 \end{aligned}$
::x=1, y2, z=4 时x2x+y+y+x2xy

$\begin{aligned} \frac{4 x y z}{y^{2} - x^{2}} \end{aligned}$ when $\begin{aligned} x = 3, y = 2, \end{aligned}$ and $\begin{aligned} z = 5 \end{aligned}$
::x=3, y=2, z=5时, 4xyzy2 - x2

$\begin{aligned} \frac{x^{2} - z^{2}}{x z - 2 x (z - x)} \end{aligned}$ when $\begin{aligned} x = - 1 \end{aligned}$ and $\begin{aligned} z = 3 \end{aligned}$
::x2-z2xz-2xx(z-x) 当 x%1 和 z=3

For 10-14, evaluate each expression using a graphing calculator.
::对于10-14,使用图形计算器对每个表达式进行评价。

$\begin{aligned} x^{2} + 2 x - x y \end{aligned}$ when $\begin{aligned} x = 250 \end{aligned}$ and $\begin{aligned} y = - 120 \end{aligned}$
::x2+2x-xy x=250 和 y 120 时

$\begin{aligned} (x y - y^{4})^{2} \end{aligned}$ when $\begin{aligned} x = 0.02 \end{aligned}$ and $\begin{aligned} y = - 0.025 \end{aligned}$
:xy-y4) 2 x=0.02 和 y 0.025 时

$\begin{aligned} \frac{x + y - z}{x y + y z + x z} \end{aligned}$ when $\begin{aligned} x = \frac{1}{2}, y = \frac{3}{2}, \end{aligned}$ and $\begin{aligned} z = - 1 \end{aligned}$
::x=12, y=32, z1时 x+y-zxy+yz+xz

$\begin{aligned} \frac{(x + y)^{2}}{4 x^{2} - y^{2}} \end{aligned}$ when $\begin{aligned} x = 3 \end{aligned}$ and $\begin{aligned} y = - 5 \end{aligned}$
:x+y) 24x2-y2 当 x=3 和 y=5 时

$\begin{aligned} \frac{(x - y)^{3}}{x^{3} - y} + \frac{(x + y)^{2}}{x + y^{4}} \end{aligned}$ when $\begin{aligned} x = 4 \end{aligned}$ and $\begin{aligned} y = - 2 \end{aligned}$
:x=4和y2时) x=3x3-y+(x+y)2x+y4

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

常规

带有分数栏的代数表达式

Expressions Involving Fraction Bars ::包含分数栏的表达式

Evaluating Expressions ::评价表达式

Evaluating Multi-Variable Expressions ::评价多变量表达式

Using a Graphing Calculator ::使用图形计算计算器

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Expressions Involving Fraction Bars
::包含分数栏的表达式

Evaluating Expressions
::评价表达式

Evaluating Multi-Variable Expressions
::评价多变量表达式

Using a Graphing Calculator
::使用图形计算计算器

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)