Course: 9.3 简化激进言论

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简化激进言论
Marco bought a square picture frame that measured 144 square inches. The wooden trim was $\begin{aligned} {1.5}^{''} \end{aligned}$ wide all the way around. How can Marco determine what size picture will fit the frame?
::马可买了一个方形图框,测量了144平方英寸。木制修剪宽1.5英寸。马科如何确定哪一幅图图与框架相匹配?

In this concept, you will learn how to simplify expressions that contain radicals.
::在这个概念中,你将学会如何简化含有激进的表达方式。

Simplifying Radical Expressions
::简化激进表达式

The square root symbol , $\begin{aligned} \sqrt{x} \end{aligned}$ , is also called a radical .
::平方根符号 x, 也被称为激进符号。

You may already know that an expression is a mathematical sentence that contains numbers and operations. Sometimes radicals are included in a sentence.
::您可能已经知道, 表达式是一个包含数字和操作的数学句。有时, 激进性会包含在句子中。

Let’s look at an example.
::让我们举个例子。

$\begin{aligned} 2 \times \sqrt{4} + 7 \end{aligned}$

This sentence says “two times the square root of four plus seven.”
::这句话写着“四加七平方根的二倍”。

First, remember the Order of Operations .
::首先,记住行动令

Parentheses
::括号括号

(including square roots )
:包括平方根)

Multiplication and Division in order from left to right
::从左到右的乘数和除法

Addition and Subtraction in order from left to right
::从左到右的加法和减法

Next, according to those rules, evaluate the square root of 4.
::其次,根据这些规则,评估4的平方根。

$\begin{aligned} \sqrt{4} = 2 \end{aligned}$

Next, substitute that value into the expression.
::接下来,将该值替换为表达式。

$\begin{aligned} 2 \times 2 + 7 \end{aligned}$

Then, complete the multiplication.
::然后完成乘法

$\begin{aligned} 2 \times 2 = 4 \end{aligned}$

And substitute that given value.
::以给定的价值代替给定的价值。

$\begin{aligned} 4 + 7 = 11 \end{aligned}$

The answer is 11.
::答案是11个

Let’s look at another example.
::让我们再看看另一个例子。

$\begin{aligned} \sqrt{4} \times \sqrt{16} - 3 \end{aligned}$

First, evaluate the radicals.
::首先,评估激进分子。

$\begin{aligned} \begin{array}{rcl} \sqrt{4} & = & 2 \\ \sqrt{16} & = & 4 \end{array} \end{aligned}$

Next, substitute these values into the expression.
::接下来,将这些值替换为表达式。

$\begin{aligned} 2 \times 4 - 3 \end{aligned}$

Next, complete the multiplication.
::下一步,完成乘法。

$\begin{aligned} 2 \times 4 = 8 \end{aligned}$

Then, complete the subtraction.
::然后完成减法

$\begin{aligned} 8 - 3 = 5 \end{aligned}$

The answer is 5.
::答案是5。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Marco and his 144-square-inch picture frame with the wooden trim.
::早些时候,你被问及马可和他144平方英寸的画框与木制修饰。

He needs to figure out what size picture he can put in.
::他需要弄清楚他能画出多大的图片

First, you know that the length of each side can be determined by taking a square root.
::首先,你知道每一方的长度可以通过一个平方根来确定。

$\begin{aligned} s = \sqrt{144} \end{aligned}$

::=144 s=144

Next, you know how much space the frame uses on each side.
::接下来,你知道框架在每一边使用多少空间。

$\begin{aligned} 1.5 + 1.5 = \end{aligned}$

Then, subtract the value of the wooden frame from the length of the side to get the side dimension for a photo.
::然后,从侧面长度中减去木框的值,以获得相片的侧边维度。

$\begin{aligned} \sqrt{144} - 3 = \end{aligned}$

Solve the radical and substitute.
::解决激进派和替代派

$\begin{aligned} 12 - 3 = 9 \end{aligned}$

The answer is 9. Marco can use a 9 inch by 9 inch photo in his new frame.
::答案是9. Marco可以使用9英寸乘9英寸的照片在他的新框中。

Example 2
::例2

Evaluate the expression.
::评估表达方式。

$\begin{aligned} 6 + \sqrt{9} - \sqrt{49} + 5 \end{aligned}$

First, solve the radicals for the square roots.
::首先,解决平方根的激进

$\begin{aligned} \begin{array}{rcl} \sqrt{9} & = & 3 \\ \sqrt{49} & = & 7 \end{array} \end{aligned}$

Next, substitute these values in the expression.
::接下来,在表达式中替换这些值。

$\begin{aligned} 6 + 3 - 7 + 5 \end{aligned}$

Then, add and subtract.
::然后补加减。

The answer is 7.
::答案是7个

Example 3
::例3

$\begin{aligned} 6 + \sqrt{81} - 10 \end{aligned}$

First, evaluate the radical.
::首先,评估激进分子。

$\begin{aligned} \sqrt{81} = 9 \end{aligned}$

Next, substitute that value in the expression.
::接下来,在表达式中替换该值。

$\begin{aligned} 6 + 9 - 10 \end{aligned}$

Then add and subtract.
::然后加减。

The answer is 5.
::答案是5。

Example 4
::例4

$\begin{aligned} \sqrt{64} \div \sqrt{4} + 13 \end{aligned}$

First, determine the square roots.
::首先,决定平方根。

$\begin{aligned} \begin{array}{rcl} \sqrt{64} & = & 8 \\ \sqrt{4} & = & 2 \end{array} \end{aligned}$

Next, substitute those values in the expression.
::接下来,用表达式中的这些值来替代这些值。

$\begin{aligned} 8 \div 2 + 13 \end{aligned}$

Then divide and substitute.
::然后,你们分舍吧,你们以物配主。

$\begin{aligned} 4 + 13 \end{aligned}$

The answer is 17.
::答案是17。

Example 5
::例5

Tiffany needs to trim a square piece of wood for her shop class. The board is 25 square inches. If Tiffany cuts off 1 inch all the way around, write the mathematical expression for the side dimensions of the new piece of wood. Then solve for the dimension.
::蒂芬尼需要修整一平方块木头来做她的商店。板子是25平方英寸。如果蒂芬尼切开1英寸左右, 请为新木头的侧边尺寸写数学表达式。然后解开维度。

First, draw a picture.
::首先,画一张图。

Next, you know that the original board is square and covers 25 square inches, so each side of the original board is
::接下来,你知道,原板是平方的, 覆盖面积25平方英寸, 所以原板的每侧都是

$\begin{aligned} old s = \sqrt{25} \end{aligned}$

::旧s=25

Then, you know that Tiffany will cut off one inch all the way around for a total of two inches subtracted from each side. The mathematical expression for the new side is:
::然后,你知道Tiffany会切断一英寸的距离, 每一边总共减掉两英寸。新边的数学表达方式是:

$\begin{aligned} new side = \sqrt{25} - 2 \end{aligned}$

::新侧面=25-2

Take the square root, then subtract.
::取平方根,然后减去。

$\begin{aligned} \begin{array}{rcl} new side & = & 5 - 2 \\ new side & = & 3 i n . \end{array} \end{aligned}$

::新面=5-2新面=3英寸。

The answer is 3 inches. The new side is 3 inches.
::答案是3英寸,新边是3英寸

Review
::回顾

Evaluate each radical expression.
::评估每个激进的表达方式。

$\begin{aligned} 2 + \sqrt{9} + 15 - 2 \end{aligned}$

$\begin{aligned} 3 \cdot 4 + \sqrt{169} \end{aligned}$

$\begin{aligned} \sqrt{16} \cdot \sqrt{25} + \sqrt{36} \end{aligned}$

$\begin{aligned} \sqrt{81} \cdot 12 + 12 \end{aligned}$

$\begin{aligned} \sqrt{36} + \sqrt{49} - \sqrt{16} \end{aligned}$

$\begin{aligned} 6 + \sqrt{36} + 25 - 2 \end{aligned}$

$\begin{aligned} 4 (5) + \sqrt{9} - 2 \end{aligned}$

$\begin{aligned} 15 + \sqrt{16} + 5 \end{aligned}$

$\begin{aligned} 3 (2) + \sqrt{25} + 10 \end{aligned}$

$\begin{aligned} 4 (7) + \sqrt{49} - 12 \end{aligned}$

$\begin{aligned} 2 (4) + \sqrt{9} - 8 \end{aligned}$

$\begin{aligned} 3 (7) + \sqrt{25} + 21 \end{aligned}$

$\begin{aligned} 8 (3) - \sqrt{36} + 15 - 2 \end{aligned}$

$\begin{aligned} 19 + \sqrt{144} - 22 \end{aligned}$

$\begin{aligned} 3 (4) + \sqrt{64} + \sqrt{25} \end{aligned}$

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。

Section outline

General

简化激进言论

Simplifying Radical Expressions ::简化激进表达式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Simplifying Radical Expressions
::简化激进表达式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)