Course: 7.2 评估激进言论和小分权

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评价激进言论和小分权
The first dance of the year at the local middle school was held on the last Friday in September. The school council decided that the price of admission into the dance would be a non-perishable food item which would be donated to the local food bank. On Monday, the members of the students’ council sorted the food items into dry goods and canned goods.
::当地中学今年的第一次舞蹈是在9月的最后一个星期五举行的。学校理事会决定舞蹈的招生价格是不可食用的食品,捐赠给当地食品银行。星期一,学生理事会成员将食品分类为干货和罐头食品。

The total number of canned goods was 321. The principle asked the students to represent 321 using a mathematical expression and the writer of the best expression would win free admission into the next dance. How does this winning expression represent 321 cans?
::罐装物品总数为321件。原则要求学生使用数学表达法代表321件,而最佳表达法的作者将赢得下一支舞蹈的自由录取。成功表达法如何代表321罐?

In this concept, you will learn to evaluate radical expressions and fractional exponents.
::在这个概念中,你将学会评价激进的表达方式和分数的表率。

Radical Expressions
::激进言论

Remember a radical consists of a radical sign, a radicand and an index. The radical sign is the symbol you use to denote square root , $\begin{aligned} \sqrt{} \end{aligned}$ , the radicand is the number under the radical sign, $\begin{aligned} \sqrt{radicand} \end{aligned}$ , and the index is the number that tells what root to find, $\begin{aligned} \sqrt[3]{radicand} \end{aligned}$ . The cube root of 27 can be expressed as $\begin{aligned} \sqrt[3]{27} \end{aligned}$ .
::记住激进的由激进符号、弧线和指数组成。激进符号是您用来表示正方根的符号 , 弧线是激进符号下的数字 , 弧线, 指数是表示要找到什么的根的数字 , 弧线 3 。 27 的立方根可以表示为 273 。

In the radical expression $\begin{aligned} \sqrt[3]{27} \end{aligned}$ the exponent of 27 is understood as being one. Therefore the radical expression could also be written as $\begin{aligned} \sqrt[3]{27^{1}} \end{aligned}$ or as $\begin{aligned} {(\sqrt[3]{27})}^{1} \end{aligned}$ . The number that raises a base to a power is called an exponent . All of the exponents that you have used so far have been whole numbers. However, an exponent can also be a fraction such that its numerator is the exponent of the base and its denominator is the index of the radical. The base of a radical expression is the radicand. This means that the radical expression $\begin{aligned} \sqrt[3]{27} \end{aligned}$ can also be written using a fractional exponent as $\begin{aligned} 27^{\frac{exponent}{index}} \end{aligned}$ or $\begin{aligned} 27^{\frac{1}{3}} \end{aligned}$ .
::在激进表达式273中,273的引言被理解为一个。因此,激进表达式也可以写为2713或27311。将一个基础提升到一个权力的数值被称为推言人。到目前为止,你使用的所有推言词都是整数。然而,一个推理也可以是一个小数,以至于其分子是基数的引言,其分母是基数的指数。激进表达式的基数是射线。这意味着,激进表达式273也可以用273或2713的分数指数来写。

The square root of 16 can written as the radical expression $\begin{aligned} \sqrt{16} \end{aligned}$ . The index ‘2’ is not written here but is understood. The exponent of ‘1’ for 16 is also not written but is understood. Therefore the $\begin{aligned} \sqrt{16} \end{aligned}$ can be written as $\begin{aligned} 16^{\frac{1}{2}} \end{aligned}$ using fractional exponents.
::16 的平方根可以写成激进表达式 16 。指数 ' 2 ' 不是在这里写,而是在这里被理解。 16 的 ` 1 ' 的缩略语也没有写,而是被理解。因此, 16 可以是使用分数表写为 1612 的 1612 。

Whether the expression is written in radical form or fractional form it can be evaluated.
::表达式是激进形式还是分形形式,都可以评价。

$\begin{aligned} 27^{\frac{1}{3}} or \sqrt[3]{27} \end{aligned}$

::2713或273

Both of these expressions mean to find the cube root of 27. The cube of 27 is the number which multiplied by itself three times equals 27.
::这两个表达式都表示要找到27的立方根,27的立方体是乘以3乘以27的数值。

$\begin{aligned} 27 = 3 \times 3 \times 3 = 3^{3} \end{aligned}$

The cube root of 27 is 3.
::27的立方根是3

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about the dance and all the canned goods. You need to figure out how $\begin{aligned} 361^{\frac{1}{2}} + 302 \end{aligned}$ equals 321 cans of food.
::早些时候,有人给了你舞蹈和罐头食品的问题。你需要弄清楚36112+302 如何等于321罐食物。

First, write the fractional power as a radical expression.
::首先,将分数力量写成激进表达方式。

$\begin{aligned} \sqrt{361} \end{aligned}$

Next, write the entire expression.
::接下来,写整个表达式。

$\begin{aligned} \sqrt{361} + 302 \end{aligned}$

Next, determine the square of 361.
::下一个,确定361的平方。

$\begin{aligned} 361 = 19 \times 19 = 19^{2} \end{aligned}$ . The square is 19.
::361=19x19=192 方块是19

Next, substitute 19 into the expression for $\begin{aligned} \sqrt{361} \end{aligned}$ .
::下面,将19改为361。

$\begin{aligned} 19 + 302 \end{aligned}$

Then, add the numbers.
::然后加上数字。

$\begin{aligned} 19 + 302 = 321 \end{aligned}$

The answer is 321.
::答案是321号

The expression represents 321 cans of food.
::这个表达方式代表了321罐食物。

Example 2
::例2

Evaluate the following fractional power.
::评估以下几分功率。

$\begin{aligned} 64^{\frac{2}{3}} \end{aligned}$

First, determine what the fractional represents. The ‘2’ is the exponent and the ‘3’ is the index.
::首先,确定分数代表什么。 " 2 " 是指数, " 3 " 是指数。

Next, rewrite the fractional power as a radical. Remember, the exponent $\begin{aligned} \frac{2}{3} \end{aligned}$ represents $\begin{aligned} \frac{exponent}{index} \end{aligned}$ .
::接下来,将分数能量重写为激进。记住, 引号 23 代表指数指数。

$\begin{aligned} {(\sqrt[3]{64})}^{2} \end{aligned}$

Next, determine the cube of 64. It is the number which multiplied by itself three times equals 64.
::接下来,确定64的立方体。数字本身乘以3倍等于64。

$\begin{aligned} 64 = 4 \times 4 \times 4 = 4^{3} \end{aligned}$ . The cube of 64 is 4.
::64=4x4x4=43。 64 的立方体是 4。

Next, write the new expression by substituting 4 for $\begin{aligned} \sqrt[3]{64} \end{aligned}$ .
::接下来,写下新表达式, 代之以 4 取代 643 。

$\begin{aligned} (4)^{2} \end{aligned}$

Then, evaluate the new expression.
::然后,评估新的表达方式。

$\begin{aligned} 4^{2} = 4 \times 4 = 16 \end{aligned}$

The answer is 16.
::答案是16岁

Example 3
::例3

Evaluate each of the following expressions.
::评估以下每一种表达方式。

$\begin{aligned} \sqrt{121} \end{aligned}$

$\begin{aligned} \sqrt[4]{16} \end{aligned}$

$\begin{aligned} 125^{\frac{2}{3}} \end{aligned}$

a) $\begin{aligned} \sqrt{121} \end{aligned}$
:a) 121

First, determine what the expression means.
::首先,确定表达方式的含义。

Find the square root of 121. The square root is the number which multiplied by itself twice equals 121.
::找出121的平方根平方根是乘以2等于121的数字

Next, determine the square of 121.
::下一步,确定121的正方形。

$\begin{aligned} 121 = 11 \times 11 = 11^{2} \end{aligned}$

Then, write the expression equal to its square.
::然后写出与其正方形相等的表达式。

$\begin{aligned} \sqrt{121} = 11 \end{aligned}$

The answer is 11.
::答案是11个

b) $\begin{aligned} \sqrt[4]{16} \end{aligned}$
::b) 164

First, determine what the expression means.
::首先,确定表达方式的含义。

Find the fourth root of 121. The fourth root is the number which multiplied by itself four times equals 16.
::找出121的第四根,第四根是乘以四倍等于16的数值。

Next, determine the fourth root of 16.
::下一步,决定16的第四根

$\begin{aligned} \begin{array}{rcl} 16 & = & 2 \times 2 = 4 \times 2 = 8 \times 2 = 16 \\ 16 & = & 2 \times 2 \times 2 \times 2 = 2^{4} \end{array} \end{aligned}$

Then, write the expression equal to its fourth root.
::然后写出等于其第四根的表达式。

$\begin{aligned} \sqrt[4]{16} = 2 \end{aligned}$

The answer is 2.
::答案是2。

c) $\begin{aligned} 125^{\frac{2}{3}} \end{aligned}$
:c) 12523

First, determine what the expression means.
::首先,确定表达方式的含义。

Find the cube root of 125 and square the cube root.
::查找 125 的立方根和立方根的正方根。

Next, write the expression to represent its meaning.
::接下来,写出表达式来表达其含义。

$\begin{aligned} {(125^{\frac{1}{3}})}^{2} \end{aligned}$

Next, perform the operation in the parenthesis. Determine the cube root of 125.
::下一步,在括号中执行操作。确定125的立方根。

$\begin{aligned} \begin{array}{rcl} 125 & = & 5 \times 5 = 25 \times 5 = 125 \\ 125 & = & 5 \times 5 \times 5 = 5^{3} \end{array} \end{aligned}$

Next, write the new expression by substituting ‘5’ for $\begin{aligned} (125^{\frac{1}{3}}) \end{aligned}$ .
::接下来,用“5”代替(12513)来写新表达式。

$\begin{aligned} (5)^{2} \end{aligned}$

Next, evaluate the new expression.
::接下来,评价新的表达式。

$\begin{aligned} (5)^{2} = 5 \times 5 = 25 \end{aligned}$

Then, write the given expression equal to the evaluated value.
::然后,写入与被评估值相等的表达式。

$\begin{aligned} {(125^{\frac{1}{3}})}^{2} = 25 \end{aligned}$

The answer is 25.
::答案是25岁

Example 4
::例4

Rewrite the following radical expression using fractional exponents.
::使用分数表征重写以下的激进表达式。

$\begin{aligned} {(\sqrt[4]{81})}^{3} \end{aligned}$

First, recall what is represented by a fractional exponent.
::首先,回顾一下一个小小的表率代表什么。

The numerator represents the exponent and the denominator represents the index. $\begin{aligned} (\frac{exponent}{index}) \end{aligned}$
::分子代表指数,分母代表指数。 (指数)

$\begin{aligned} {(\sqrt[index \to 4]{81})}^{3 \leftarrow exponent} \end{aligned}$

:81指数__________________________________________________________________________________________________________

Next, write the base (radicand) with the fractional exponent.
::接下来,用分数表写底部(radicand) 。

$\begin{aligned} 81^{\frac{3}{4}} \end{aligned}$

Then, write the radical expression equal to the fractional power.
::然后写出与分数力相等的激进表达式。

$\begin{aligned} (\sqrt[4]{81})^{3} = 81^{\frac{3}{4}} \end{aligned}$

The answer is $\begin{aligned} 81^{\frac{3}{4}} \end{aligned}$ .
::答案是8134。

Example 5
::例5

Rewrite the following expression as a radical.
::将以下表达式重写为激进表达式。

$\begin{aligned} 32^{\frac{3}{5}} \end{aligned}$

First, recall what the fractional exponent represents.
::首先,请记住分数表代表什么。

The fractional exponent represents $\begin{aligned} (\frac{exponent}{index}) \end{aligned}$ .
::分数前列数表示( Exponent index) 。

Next, write the base of 32 as the radicand of a radical expression.
::接下来,将32的基数写成激进表达式的射线。

$\begin{aligned} \sqrt{32} \end{aligned}$

Next, write the index in its correct position to indicate what root of 32 needs to be found.
::接下来,在正确的位置写索引,以表明需要找到32根的根。

$\begin{aligned} \sqrt[5]{32} \end{aligned}$

Next, write the exponent in its correct position to indicate that the fifth root of 32 must be raised to the power of 3.
::接下来,写出正确位置的推手表示32的第五根必须提升到三根

$\begin{aligned} {(\sqrt[5]{32})}^{3} \end{aligned}$

Then, write the fractional power equal to the radical expression.
::然后写出与激进表达方式相等的分数权力。

$\begin{aligned} 32^{\frac{3}{5}} = {(\sqrt[5]{32})}^{3} \end{aligned}$

The answer is $\begin{aligned} {(\sqrt[5]{32})}^{3} \end{aligned}$ .
::答案是(325)3。

Review
::回顾

Evaluate each fractional power.
::评估每个分数功率。

$\begin{aligned} 64^{\frac{1}{2}} \end{aligned}$

$\begin{aligned} 16^{\frac{1}{2}} \end{aligned}$

$\begin{aligned} 144^{\frac{1}{2}} \end{aligned}$

$\begin{aligned} 81^{\frac{1}{2}} \end{aligned}$

$\begin{aligned} 9^{\frac{1}{2}} \end{aligned}$

$\begin{aligned} 25^{\frac{1}{2}} \end{aligned}$

$\begin{aligned} 216^{\frac{1}{3}} \end{aligned}$

$\begin{aligned} 100^{\frac{1}{2}} \end{aligned}$

$\begin{aligned} 16^{\frac{1}{4}} \end{aligned}$

$\begin{aligned} 256^{\frac{1}{4}} \end{aligned}$

$\begin{aligned} 125^{\frac{1}{3}} \end{aligned}$

$\begin{aligned} 36^{\frac{1}{2}} \end{aligned}$

$\begin{aligned} 81^{\frac{1}{4}} \end{aligned}$

$\begin{aligned} 121^{\frac{1}{2}} \end{aligned}$

$\begin{aligned} 169^{\frac{1}{2}} \end{aligned}$

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

评价激进言论和小分权

Radical Expressions ::激进言论

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)

Radical Expressions
::激进言论

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)