7 Roots and Radical Expressions7根与根的表达式-PPT精选文档
Algebra1中英版对照目录
分步和混合运算解一元一次方程 P92—P99 2—4：Solving Equations with Variables on Both Sides
变量在两边的一元一次方程的解法 P100—P106 2—5：Solving for a Variable 求变量的值（一元一次）P107—P113
指数幂的乘法运算（同底数幂相乘和积的乘方运算）P460—P466 7—4：Division Properties of Exponents 指数幂的除法 P467—P474 Quiz for Lessons 7-1 through 7-4 第七章 1—4 课小测试 P474—P475 7—5：Polynomials 多项式 P476—P483 7—6：Adding and Subtracting Polynomials 多项式的加减运算 P484—P491 7—7：Multiplying Polynomials 多项式的乘法（单项式与多项式相乘、多项
Algebra 1 与中文数学课本内容在顺序上的对照
Algebra 1
中教数学
CHAPTER 0---- To The Student 0—1：Geometry Formulas
几何公式（s quare，rec tangle,triangle,c irc le）Z3— Z4 0—2：Tree Diagrams 树状图 Z4—Z6 0—3：The Coordinate Plane 直角坐标系 Z7—Z8 0—4：Rounding and Estimating 四舍五入 Z9—Z11 0—5：Adding and Subtracting Decimals 小数的加减运算 Z12—Z13 0—6：Multiplying and Dividing Decimals 小数的乘除运算 Z14—Z16 0—7：Prime and Composite Numbers 素数与合数 Z17—Z18 0—8：Factoring 因数 Z19—Z20 0—9：GCF and LCM 最大公约数和最小公倍数 Z21—Z22
Simplifying Radical Expressions
EXAMPLES:
2 −2 x x = 2 = −1 2 4 x 2
1 1 1 81 = 1 = 4 = 81 4 81 3 −3 1 1 1 4 16 = 3 = = 16 4 4 (16)3 8
−1 4
Simplifying Radicals by using the Product Rule
10 m 3 ⋅ 6 5 m 2 = 7⋅ 5 =
3
50 m 5
7⋅ 5
*This one can not be simplified any further due to their indexes (2 and 3) being different!
Simplifying Radicals involving Variables:
(16 ) ( −2 ) = ( −2 )
1
16
2
5
3
power root
Negative Exponents:
Remember that a negative in the exponent does not make the number negative! If a base has a negative exponent, that indicates it is in the “wrong” position in fraction. That base can be moved across the fraction bar and given a postive exponent.
a & m b are real If numbers and m is a natural number, then
根的名词解释
根的名词解释在我们日常生活中，我们经常会听到“根”这个词。
无论是在科学、生物、植物、语言或者数学上，根都是一个非常重要的概念。
然而，对于普通人来说，根的含义可能并不是那么明确。
本文将从不同的角度来解释“根”的含义。
第一部分：根的生物学含义在生物学中，“根”是指植物的根系。
植物的根是它们生命的基石，起着吸收水分、养分和提供支撑的重要作用。
植物的根通常分为主根和侧根，它们通过顶端的根尖进行生长。
根毛是根的一部分，它们能够增加根的表面积，从而提高水分和养分的吸收能力。
植物的根系形态不同，有些植物的根系呈现出扁平的形状，有些植物的根系则非常深入土壤。
而且，根还可以通过地下茎和根茎繁殖。
第二部分：根的数学含义在数学中，根是一个常见的概念。
根是数学方程中使方程等式成立的未知数的值。
例如，对于方程x^2 - 4 = 0，根就是使方程成立的x的值。
一个方程可能有多个根，也可能没有根。
根的概念在代数学和数论中非常重要，它们帮助我们解决各种数学问题。
第三部分：根的语言含义在语言中，“根”是指一个词的核心部分，通常具有基本的含义。
它是构成一个单词的基本元素，能够用于构造相关的词汇。
例如，词根“电”在很多词汇中都有出现，如“电视”、“电脑”、“电子”等等。
通过词根，我们可以推测一个词的含义或者联想到其他相关概念。
第四部分：根的隐喻含义除了上述的生物学、数学和语言含义之外，根还有一些隐喻含义。
在人类文化中，根常常被用来表示血缘关系、传承和渊源。
例如，“家族之根”表示一个家族的起源和血统。
在艺术、文学作品中，根也常被用来象征人类无法割舍的身份和归属感。
综上所述，根在不同的领域中具有不同的含义。
从生物学、数学、语言到文化，根都是一个重要的概念。
它代表着生命的基础、数学方程的解、词汇的核心以及文化的渊源。
通过理解根的含义，我们能够更好地理解生活和世界的本质。
7 Roots and Radical Expressions7根与根的表达式
index
Radical sign
na
radicand
How to find the root (other than a square root), using a graphing calculator
1. Input the root you are going to take (for example, if you are taking the third root of a number, start with the 3).
radicals • Complex numbers • Conjugates
What is a monomial?
An expression that is a number, that may or may not include a variable.
MONOMIALS
10 xy
2x2
4x
NOT MONOMIALS
4
x8
x
8 1
5x
Real Roots
• Real roots are the
possible solutions to a
number, raised to a power.
24 (2)4 16 soboth2andnegativ2eare
x2 xx
rootosf16tothefourtphower
Let’s try some
Simplify each expression. Use the absolute value symbols when needed.
4x2 y4
3 27x6y
Solutions
Simplify each expression. Use the absolute value symbols when needed.
生物中根和叶的英语
生物中根和叶的英语Roots and Leaves in BiologyRoots and leaves are two essential components of plants that play crucial roles in their growth, development, and survival. These two structures work together to ensure the plant's overall health and function, each contributing to the plant's ability to thrive in its environment.Roots are the underground part of the plant that are responsible for anchoring the plant in the soil and absorbing water and nutrients. They are typically composed of a main root, known as the taproot, and a network of smaller roots called lateral roots. The root system is essential for the plant's survival as it provides the necessary resources for growth and development.The primary function of roots is to absorb water and minerals from the soil. The root system is equipped with specialized structures called root hairs, which greatly increase the surface area of the roots, allowing for more efficient absorption of these essential resources. The water and minerals absorbed by the roots are then transported to the leaves and other parts of the plant through a complex systemof vascular tissues.In addition to their role in nutrient and water absorption, roots also play a crucial role in anchoring the plant in the soil. The root system provides stability and support, preventing the plant from being uprooted by wind, rain, or other environmental factors. This anchoring function is particularly important for taller plants, such as trees, which must withstand the forces of nature.Leaves, on the other hand, are the aboveground part of the plant that are responsible for photosynthesis, the process by which plants convert light energy from the sun into chemical energy in the form of glucose. Leaves are typically green in color due to the presence of a pigment called chlorophyll, which is essential for the photosynthesis process.The structure of leaves is highly specialized to facilitate photosynthesis. They are composed of a flat, thin blade, known as the lamina, which is supported by a network of veins. The veins transport the water and minerals from the roots to the leaves, as well as the glucose produced during photosynthesis to the rest of the plant.The surface of the leaves is also covered with tiny pores called stomata, which allow the exchange of gases between the plant andthe atmosphere. During photosynthesis, the stomata open to allow carbon dioxide to enter the leaf, while oxygen and water vapor are released as byproducts.In addition to their role in photosynthesis, leaves also play a crucial role in the plant's water regulation. Transpiration, the process by which water evaporates from the leaves, helps to draw water up from the roots, creating a continuous flow of water through the plant. This process is essential for the plant's overall health and growth.The relationship between roots and leaves is a symbiotic one, with each structure relying on the other for its survival and function. The roots provide the leaves with the necessary water and minerals, while the leaves produce the glucose that the roots need for growth and development. This interdependence is essential for the overall health and well-being of the plant.In conclusion, roots and leaves are two essential components of plants that work together to ensure the plant's survival and growth. Roots are responsible for anchoring the plant and absorbing water and nutrients, while leaves are responsible for photosynthesis and water regulation. The relationship between these two structures is a symbiotic one, with each relying on the other for its function and survival.。
作文一帆风顺的根和茎
作文一帆风顺的根和茎英文回答：The root and stem of a plant are essential parts that contribute to its growth and overall health. The root serves as an anchor, providing stability and support to the plant. It also absorbs water and nutrients from the soil, which are necessary for the plant's survival. Without a strong and healthy root system, a plant would struggle to grow and thrive.Similarly, the stem plays a crucial role in a plant's life. It acts as a transportation system, delivering water and nutrients from the roots to the leaves and other parts of the plant. The stem also provides structural support, allowing the plant to stand upright and reach towards the sunlight. Additionally, the stem contains vascular tissues that help in the movement of water, nutrients, and sugars throughout the plant.In Chinese, we often use the phrase "根深蒂固" (gēnshēn dì gù) to describe something that has a strong foundation or is deeply rooted. This phrase can be used to emphasize the importance of a solid root system in plants. Similarly, we use the idiom "枝繁叶茂" (zhī fán yè mào) to describe a plant with a lush and thriving appearance. This idiom highlights the role of the stem in supporting the growth and development of a plant.中文回答：植物的根和茎是促进其生长和整体健康的重要部分。
radical form数学
radical form数学在数学中，"radical form"是指一种特殊的数学表达方式，用来表示一个数的平方根，立方根或更高次方根。
它通常以符号√表示，被称为根号。
在这种表达方式中，数被写成一个根号符号下的一个数，代表它的根。
在数学中，我们通常使用10进制或二进制来表示数字。
我们可以通过有限的数字和运算符（如加减乘除）来表示任意数量的数字。
然而，有些数无法用这些常见的表示方法来准确表示。
例如，无理数π（圆周率）和√2（根号2）是两个常见的例子。
由于它们是无限不循环的小数，无法准确表示为有限的十进制数。
在数学中，根号的应用非常广泛。
它常被用来求解方程，找到未知数的值。
例如，我们可以使用根号来解决二次方程。
在二次方程ax^2 + bx + c = 0中，我们可以使用根号√来得到x的值：x = (-b ±√(b^2 - 4ac)) / (2a)。
这个公式叫做二次方程根的公式。
它是由一种基本的数学方法推导出来的。
通过将不等式转化为等式，然后使用根号表示根的方式来解决方程。
根号还可以表示一个数的平方根。
例如√4 = 2，√9 = 3。
这些数都是正整数，可以被准确地表示为根号。
然而，有些数的平方根无法被准确地表示为有限的一个数。
例如，√2是一个无理数，它无法用有限的数字表达。
在这种情况下，我们将√2称为根号的形式，√2即为√2本身。
根号的形式也可以用来表示立方根和更高次方根。
例如，∛8 = 2，∛27 = 3。
这些数都是正整数，可以被准确地表示为根号的形式。
然而，对于一些数，立方根和更高次方根无法被准确地表示为有限的数字。
在这种情况下，我们将它们称为根号的形式，即将它们表示为一个根号符号下的一个数。
根号的形式在数学中有很多应用。
它们可以用来计算复杂函数的值，求解高次方程等。
在几何学中，根号的形式可以用来计算图形的尺寸和面积。
在物理学中，根号的形式可以用来计算物体的质量、速度、加速度等。
radical form数学
Radical Form数学一、概述Radical Form是数学中的一个重要概念，它在代数运算和方程求解中发挥着重要作用。
本文将围绕radical form展开详细讨论，包括其定义、性质和应用。
二、radical form的定义1. radical form的概念在代数学中，radical form是指一个数或量的开平方的代数表达式，一般形式为√n，其中n为被开方数，且n为非负实数。
√9表示被开方数为9的根号表达式，其结果为3。
2. radical form的特点(1) radical form的指数必须为奇数在radical form中，被开方数的指数必须为奇数。
如果指数为偶数，那么开方的结果将是正数，而无法表达负数的情况。
radical form可以用根号符号表示，也可以用指数形式表示。
√9可以表示为9的1/2次方，即9^(1/2)，这两种表示形式是等价的。
三、radical form的运算1. radical form的加法和减法(1) 加法：对于两个radical form，其被开方数相同，可以直接进行加法。
√3+√3=2√3。
(2) 减法：同样地，对于两个radical form，其被开方数相同，可以直接进行减法。
√5-√2=√5-√2。
2. radical form的乘法和除法(1) 乘法：对于两个radical form，可以利用乘法的交换律和结合律进行合并。
√2*√3=√(2*3)=√6。
(2) 除法：在进行radical form的除法时，要先化简被开方数的因数，然后再进行开方。
√8/√2=√(8/2)=√4=2。
1. radical form在方程求解中的应用(1) 一次方程：对于一次方程，如果方程中含有radical form，可以利用开平方的性质将其转化为简单的算式，进而求解方程。
(2) 二次方程：在二次方程中，如果无法直接进行因式分解，那么可以利用radical form的概念，通过开方运算将方程转化为标准的二次方程求解。
radical用法
radical用法一、Radical的基本含义radical是一个形容词和名词，主要表示“根本的”、“基本的”、“激进的”、“极端的”，也可表示“极有造诣的”、“彻底的”、“首创的”，还可表示“基底”、“骨干”。
二、Radical的词义辨析fundamental，basic，essential，principle，radical这些形容词均含“基本的”之意。
fundamental：侧重指某学科、某理论、某事物的基础。
basic：指某一科学或技术的基础或根本方法，也指某一具体事物的基础或根本属性。
essential：指事物必不可少的组成部分或主要特征。
有时特指生命体或人最必需的养分或要素。
principle：侧重指原则、原理或根本理论。
常用于政治、哲学方面。
radical：侧重指事物的根本改变或改革，有时也指事物的根源。
三、Radical的固定搭配1. radical solution：激进解决办法；2. radical view：激进观点；3. radical change：根本变化；4. radical reform：彻底改革；5. radical opinion：激进观点；6. radicalism：激进主义；7. right-wing radicalism：右翼激进主义。
1. radical用作形容词时，意思是“根本的”“基本的”，在句中可用作定语或表语。
用作名词时，意思是“激进派”，是可数名词。
2. be radical in意思是“在…方面很激进”，其中的in可以省略。
Radical后有时可接抽象名词，意为“彻底地做某事”。
Radical也可用作副词，意思是“根本地”“基础地”。
3. at/from the roots of things/problems是固定搭配，意为“事物的根源/问题的根本原因”。
at the roots of something是介词短语做状语，常置于句首，有时也可置于句末做后置定语。
七缀集 引用 -回复
七缀集引用-回复什么是七缀集？七缀集是一种数学概念，用于描述一组元素之间的关系。
它是集合论的一个分支，在数学和计算机科学中经常被使用。
在七缀集中，元素被称为项（item），这些项之间存在一种被称为七缀关系（suffix relation）的关联。
七缀关系是指，如果集合中的某个项x是另一个项y的后缀，那么x是y 的一个七缀。
七缀集的核心概念是关注元素之间的顺序关系，通过七缀关系来判断元素之间的联系。
首先，七缀集的定义就是确定一组集合中的元素，以及它们之间的七缀关系。
例如，假设有一个集合S={abc, abcd, bc, a}，其中的项abc是项abcd 的后缀，因此abc是abcd的一个七缀。
七缀集的使用可以帮助我们在集合中进行关联分析和模式识别。
七缀集可以被应用于多个领域，比如文本处理、数据挖掘、自然语言处理等。
例如，在文本处理中，可以利用七缀集来分析文章中的关键词之间的关系，从而提取出文章的主题或者进行句法分析。
七缀集还可以用于模式匹配和字符串匹配问题。
通过构建七缀集，我们可以识别字符串中的模式，从而实现字符串的查找和替换操作。
例如，在搜索引擎中，可以利用七缀集来加速关键词的匹配过程，提高搜索的效率。
为了构建七缀集，我们需要首先确定集合中的项，然后确定它们之间的七缀关系。
一种常用的方法是使用后缀树（suffix tree）来表示七缀集。
后缀树是一种数据结构，可以高效地存储和检索字符串的所有后缀。
通过构建后缀树，我们可以方便地提取出字符串的七缀集。
七缀集的应用还有很多，比如在图像处理中，可以利用七缀集来识别图像中的物体或者进行图像的特征提取。
另外，在自然语言处理中，七缀集可以被用来进行词性标注、语义分析和命名实体识别等任务。
总结起来，七缀集是一种用于描述元素之间关系的数学概念，可以应用于集合论、文本处理、数据挖掘、模式匹配以及其他领域。
通过构建七缀集，我们可以提取出元素之间的七缀关系，从而实现关联分析和模式识别，并且提高数据处理的效率。
2023秋济南版八上第五节《根的结构与功能》ppt4
A 保留水分和无机盐 B 避免伤害分生区 C 避免伤害根冠 D 避免伤害根毛和幼根
5、根生长最快的部分是 （ C ）
A 根冠
B 分生区
C 伸长区 D 成熟区
6.根吸收水分和无机盐的主要部分是( D )
A.主根根尖的根冠部分 B.侧根根尖的伸长区
C.须根根尖的成熟区 D.各种根尖的根毛区
7.在成熟区内部一些上下连接细胞的横壁消失形成 中空的管道,叫做( 导管),通过它向上
测试题；
1、由种子的胚根直接发育成的根叫（ B ）
A 直根 B 主根
C 侧根 D 不定根
2、根中的导管，开始分化形成的部位是 （ C ）
A 分生区 B 伸长区
C 成熟区 D 成熟区的上部
3、根适于吸水的特点
（A ）
A 成熟区有大量的根毛 B 有分生区
C 有伸长区
D 有根冠
4、移苗时根部带土是为了
特征
功能
根冠 分生区 伸长区
细胞较大,排列不整齐
保护组织
细胞较小,排列紧密 壁薄、核大、质浓
分生组织
细胞出现液泡, 迅速 伸长
保护
分裂增生
伸长（部分 吸收功能）
成熟区
停止伸长，开始分化， 形成导管和根毛
输导组织
吸收水分 和无机盐的 主要部位
关于导管：
1、死细胞 2、导管从根的成熟区上部形成，茎叶 里都有，并且相通。 3、根吸收的水和无机盐由导管向上 送到茎和叶。 4、属于输导组织。
运输( 水分)和(
) 无机盐
8. 根的长度能不断增加的原因是
什么？ 分生区的细胞不断分裂、分化和伸长区 的细胞不断伸长的缘故.
3、探究根长长的主要部位
Roots (美国小说--根)
Roots: The Saga of an American Family(First edition cover)Roots: The Saga of an American Family is a novel written by Alex Haley and first published in 1976. It was adapted into a hugely popular, 12-hour television miniseries, also called Roots, in 1977, and a 14-hour sequel, Roots: The Next Generations, in 1979.Plot introductionBrought up on the stories of his elderly female relatives -- including his Grandmother Cynthia, who was emancipated from slavery with her family in 1865 -- Alex Haley purported to have traced his family history back to “the African,” Kunta Kinte, captured by slave traders in 1767. For generations, each of Kunta’s enslaved descendants passed down an oral history of Kunta’s experiences as a free man in Gambia, along with the African words he taught them. Haley researched African village customs, slave-trading and the history of Blacks in America -- as well as made a personal visit to the griot (oral historian) of his ancestor’s African village -- to produce this colorful rendering of his family’s history from the mid-eighteenth century through the mid-twentieth century.Plot summaryThe action begins with the birth of Kunta Kinte in 1750 to a Mandinka tribesman in the village of Juffure, The Gambia. The author liberally uses many African words to describe the everyday life of this Muslim community, which sees young boys like Kunta being groomed to manhood with lessons of hunting, protecting their families, and subscribing to codes of honor under the strict supervision of village elders.Several years later, Kunta hears vague talk about “toubob” (white people) who have been spotted in the jungles nearby. Tribesmen are disappearing from other villages, never to be seen again. At the age of 16, while Kunta is on sentry duty andlooking for wood with which to fashion a drum, he is ambushed by four slave catchers. Although he fights back, he is no match for them, and is chained and hauled off to a ship for the beginning of a horrifying sea voyage. On the journey, he finds that the Mandinka warrior, Kintango, who trained him to manhood has also been captured. Kintango provides Kunta with much verbal support. Kintango later dies in the revolt. Chained to each other and to their beds in the dark, dank hold, the slaves lie in their own excrement and become violently ill. Once or twice a week, the whites bring them up to the deck in chains in order to clean the hold. On one such occasion, the slaves, who have managed to communicate with each other despite the many different languages they speak, conspire to overthrow the whites. The revolt is quashed by the white sailors, but an outbreak of vomiting, fever, and diarrhea wipes out one third of the Black captives and half of the whites. This attrition rate was typical for slave ships of the time.At a slave auction, Kunta is bought for $850 by John Waller of Spotsylvania County, Virginia. Given the name “Toby” and assigned to work as a field laborer on Massa Waller’s plantation, Kunta attempts to escape four times over the next four years and is punished, each time more severely than the last. Unlike the American-born blacks on the plantation, who have not been taught to read or write and are treated more like children than adults, Kunta can read, write, and speak fluent Arabic, and is angered by his forced enslavement. On his fourth escape attempt, slave catchers chop off half of Kunta’s right foot so that he cannot escape anymore. Incensed by the attack, John Waller’s brother, Dr. William Waller, buys Kunta from his brother and allows him to be nursed back to health by his “big house” cook, Belle.A warmhearted, American-born slave, Belle patiently nurtures a relationship with the tall, brooding African at the same time.Kunta works for several years as Dr. Waller’s gardener and later his wagon-driver before he finally plucks up the courage to ask Belle to marry him. She does, and the two have a baby at a rather advanced age. Kunta insists that the child be named Kizzy, an African name, rather than Mary, the name Belle would have preferred.In Kizzy, Kunta invests all his efforts to remind his daughter that she is the scion of a proud, free people. He teaches her many African words and patiently repeats to her the story of his capture and sale, a story that she will pass down to his grandchildren and great-grandchildren in turn.At the age of six, Kizzy becomes good friends with Dr. Waller’s niece, Missy -- to her parents’ dismay. As time goes on, Kizzy grows less close to her parents and more attached to Missy, who treats her as her personal plaything. Through Missy, Kizzy also learns how to read. This proves to be her undoing, for ten years later, Kizzy falls in love with a male slave from the plantation. When she confides to him that she knows how to read and write, he implores her to forge papers for him so he can escape, and she does. The following day, soldiers who have caught, tortured and killed the runaway slave come to Dr. Waller’s plantation and wrench Kizzy away from her parents. She is dragged away to a slave auction, never to see her parents again. Kizzy is auctioned off to a disreputable slave owner in North Carolina named TomLea. On Kizzy’s first night at his plantation, her drunken master makes crude sexual advances to her. When she refuses to have sex with him, he brutally rapes her in the barn and then drops a quarter in a jar next to her bed as thanks for her services. He continues to abuse her several times a week, leaving her a coin each time, until she is five months pregnant. She gives birth to a son whom her master insists on giving a European name, not an African one. The baby is named George.When George is born, Kizzy, who is only 17, is horrified to see that his skin is light-colored, not ebony black like her own. Her shame is intense. The other slaves at the Lea plantation advise her to forget about the father, although Massa Lea continues to visit her frequently at night. The continual abuse drives Kizzy to depression. But when Massa Lea finally leaves her alone two years later, Kizzy bonds to the other slaves and tends to her son as lovingly as she would a child born to her out of love rather than rape.George is raised like a typical field hand. In his spare time, he enjoys hanging around the gamecock pen and Uncle Mingo, the gamecock raiser, who brings in a tidy sum for Massa Lea each year in cockfighting revenues. George instantly takes an attraction to the fighting roosters because of their noble stature. Later he becomes apprenticed to Uncle Mingo and proves himself a quick learner in feeding, capturing, cleaning, and fighting gamecocks, earning himself the nickname, “Chicken George.”After George starts full-time rooster duty, there is a noticeable improvement in Massa Lea’s winnings. Chicken George attends his first cockfight at the age of 15. As the years pass, he continues to go to tournaments and backyard fights, wins money, and saves it in order to buy freedom for himself and his family. He and Massa Lea become very close. Much of the time, Massa treats Chicken George like a partner, not as a slave, thanks to the latter’s skill with the gamecocks.At the age of 18, Chicken George encourages Massa Lea to buy a slave girl named Matilda so George can marry her. Matilda gives birth to a large family of eight children. Massa Lea loses a bet with an English cockfigher, who has him pay by giving him Chicken George. After Chicken George is sent away Matilda and her children are sold to a kind couple to pay Massa Lea’s debts.In the meantime, Chicken George’s third son, Tom, a blacksmith, marries a half-Native American slave girl named Irene. The youngest of their eight children is Cynthia, Alex Haley’s grandmother.Bolstered by Massa Lea’s promise that he will receive his freedom when he returns, Chicken George comes back and gets his certificate of freedom from Massa Lea. Kizzy and one of their old slave friends are dead by the time he returns. Chicken George finds his family, but he must escape to Canada to preserve his freedom. After the American Civil War, Chicken George and his family are reunited and they move to Tennessee to start a new life as free men and women, continuing to share the stories that their great-ancestor Kunta Kinte had his daughter commit to memory so many years before.。
2019年-86 Radical Expressions and Radical Functions86根的表达式与基函数-PPT精选文档
4 Find the inverse of a y = 3 + x - 2x2. Then graph the
function and the inverse together.
y = 3 + x - 2x2
1. Interchange x and y.
x = 3 + y - 2y2
2. Solve the new equation for y.
The number or expression under a radical symbol.
square root
A number, , that when multiplied by itself produces the given Macintosh PICT im age form at is not supported
What 1. Analyze the graphs of radical functions, and evaluate
radical expressions.
2. Find the inverse of a quadratic function.
Why Radical functions are used to model many real-world relationships. For example, the relationship between the length of a pendulum and the time it takes to complete one full swing is described by a radical function.
is n o t su p p o rted
M a c in to s h P IC T im a g e fo rm a t
七子表达式翻译
七子表达式翻译（原创版1篇）目录（篇1）1.七子表达式的定义与用途2.七子表达式的翻译方法3.七子表达式翻译的实际应用正文（篇1）七子表达式翻译是指将中文文本中的七个子表达式（主语、谓语、宾语、定语、状语、补语、宾补）转换成英文表达方式。
七子表达式是中文语法的基本组成部分，掌握它们对于理解和翻译中文文本具有重要意义。
一、七子表达式的定义与用途1.主语：表示动作或状态的主体，通常是名词或代词。
例如：“我”、“你”、“他”等。
2.谓语：表示主语的动作或状态，通常是动词。
例如：“吃”、“喝”、“走”等。
3.宾语：表示动作的对象，通常是名词或代词。
例如：“饭”、“水”、“书”等。
4.定语：修饰名词，表示名词的属性。
通常由形容词、数词、量词等构成。
例如：“美丽的”、“三个”、“红色的”等。
5.状语：修饰动词、形容词、副词或整个句子，表示动作或状态的方式、地点、时间等。
例如：“在公园里”、“昨天”、“很快”等。
6.补语：补充说明宾语的动作或状态，通常由形容词、动词、介词短语等构成。
例如：“吃饱了”、“喝醉了”、“走累了”等。
7.宾补：补充说明宾语的成分，通常由形容词、名词、动词等构成。
例如：“给我”、“让他”、“找书”等。
二、七子表达式的翻译方法在翻译过程中，需要将中文的七子表达式转换成英文表达方式。
以下是一些基本的翻译方法：1.主语：通常翻译成英文的主语，例如：“I”、“you”、“he”等。
2.谓语：翻译成英文的动词，例如：“eat”、“drink”、“walk”等。
3.宾语：翻译成英文的宾语，例如：“food”、“water”、“book”等。
4.定语：翻译成英文的形容词或副词，例如：“beautiful”、“three”、“red”等。
5.状语：翻译成英文的副词、介词短语或从句，例如：“in the park”、“yesterday”、“quickly”等。
6.补语：翻译成英文的形容词、动词或介词短语，例如：“full”、“drunk”、“tired”等。
radical form数学
radical form数学摘要：1.引言2.解析radical form 的含义3.radical form 在数学中的应用4.结论正文：【引言】数学是我们日常生活中不可或缺的一部分，它为我们解决实际问题提供了强大的工具。
在数学中，有一个概念叫做radical form，它广泛应用于各个数学领域。
本文将探讨radical form 的含义以及其在数学中的应用。
【解析radical form 的含义】Radical form，中文可译为“根式”，是一种数学表达式，表示一个数的平方根。
在代数学中，我们通常用符号√来表示平方根，而用符号②表示平方。
例如，√9 表示9 的平方根，②9 则表示9 的平方。
在更广义的数学领域中，radical form 可以表示任意一个数的n 次方根，其中n 是正整数。
【radical form 在数学中的应用】Radical form 在数学中有着广泛的应用，以下是一些典型的例子：1.解方程：在代数学中，我们常常需要解决形如x=a 的方程。
通过引入radical form，我们可以将这类方程转化为求解x=√a 的形式，从而更方便地求得解。
2.计算面积和体积：在几何学中，计算图形的面积和体积常常涉及到radical form。
例如，计算一个圆的面积，我们可以用公式S=πr，其中r 表示圆的半径，π表示圆周率，②表示乘方。
3.微积分：在微积分中，radical form 也扮演着重要角色。
例如，求解函数的导数和微分，就需要用到radical form 的概念。
【结论】总之，radical form 作为数学中的一个基本概念，它在各个数学领域中都有着广泛的应用。
学海拾贝：深挖，从7个根号谈思维的养成
学海拾贝：深挖，从7个根号谈思维的养成在网上看到一个有趣的数学竞赛题目，已经在此文的姊妹篇里面给了一份解答过程（可到主页查找学海拾贝：快看，7个根号的数学竞赛题）。
不过做数学题目最崩溃的就是：看答案我理解，但让我做做不了。
曾有此感的小朋友的可以举个爪。
在这篇文章里面，我们就来说说这个问题。
原题如下：原始题目题目给了一个7层根号的方程式，让求x的整数部分。
直觉告诉我们，上去就解方程肯定是不行的：x的确切值即便可以表达出来，也是一个比较复杂的“无理式”的形式。
从头琢磨观察题目中给的这个式子，或者尝试手写一遍，可以强烈地感受到式子的“构造”规律。
以自变量x为起点，开根号；加x再开根号；加x再开根号；加x再开根号；以此类推。
换言之，这是一个嵌套形式的式子，就好像下面这个加工流程一样。
这个式子有什么变化趋势呢？对每层操作来说，上次结果加x再开根号，得到的新结果肯定比上次结果大。
但长久来看，后面的每层操作增量越来越小。
也就是说，层层运算下去，值越来越大，但也越来越稳定。
我喜欢举例说明。
我们尝试给一个x，看看规律。
比方说x=100。
我们发现，后面的计算总是根号下的（100加上10点多），值的变化越来越慢。
想的极端一点，不要说给我7层根号，就是700层根号，结果也差不多。
（说到这里，我倒是想用两层根号去蒙一蒙了，哈哈哈哈。
可以想一想，为啥用一层根号不能蒙）举例的这个100当然没什么特殊性，x=500, x=1000，规律也是一样。
琢磨式子琢磨到这个程度，七窍就通了三窍了。
以上这些写到纸上麻烦，但只在脑子里活动的话就知道其实比较流畅，基本上不动笔，属于开窍阶段。
风暴攻关然后呢？脑子又快不够用了。
做过工程的人可能都有这样一个概念，观测一个数据的时候，减去偏置就可以使被观测的量更加凸显出来。
在7层根号这个题目中，以上面图示来讲，尽管每层结果是10点多，但原始给定值的本身等于100（对于原题目，x显然与2010的平方同一数量级）。
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ex :
4
16 x y( 2 )x ( y ) 2 x y
48 4 44 24
Note: Absolute value symbols ensure that the root is positive when x is negative. They are not needed for y because y2 is never negative.
Let’s try some
Simplify each expression. Use the absolute value symbols when needed.
4x y
2
4
3
27x y
6
Solutions
Simplify each expression. Use the absolute value symbols when needed.
What is a monomial?
An expression that is a number, that may or may not include a variable.
MONOMIALS
NOT MONOMIALS
2x
2
10xy
4x
4 x
x8
8
1
5x
Real Roots
• Real roots are the possible solutions to a number, raised to a 24 (2)4 16 power.
power
Vocabulary and Properties
Radical sign
index
n
a
radicand
How to find the root (other than a square root), using a graphing calculator
1. Input the root you are going to take (for example, if you are taking the third root of a number, start with the 3). 2. Press MATH and select option 5 x 3. Enter the value you are taking the root of. Ex: 4 81 4 MATH 5 81 ENTER
3
Practice: Find each root
3
10648
5
16807
Solutions: 22, 7, and
ERR: NONREAL ANS
Let’s take a closer lookies and Notation:
n
aa
When n is an even number
Why? We want to make sure that the root is always positive when the index is an even number
Here's another wa y tolookat it : IF x 5, x2 52 25 5 x BUT if x 5, x2 (5)2 25 5 x so we want to makesure that x is alwaysa positive with evenindexesso thatit is truebackwards andforwards.Weuse theabsolutevaluetomakesure it stays positive.
2
ex :
Notice that the index is an odd number here . . .
3
xy x( y) x y
3 6 3 3 23
2
Absolute value symbols must not be used here. If x is negative, then the radicand is negative and the root must also be negative.
4x y
2
4
3
27x y
6
Properties of Exponents – let’s review . . .
NEGATIVE EXPONENT RULE
5
a
2
1 2 5
n
1 n a
PRODUCT OR POWER RULE
2 2 2
10 20
30
a a a
m n
x xx or x x so both x and - x are roots of x to the second
2
s oboth 2and negative 2are roots of16tothe fourth power
5 3 125 so - 5 is the only possible real root of - 125 (why can' t it be 5?)
7. Roots and Radical Expressions
In this chapter, you will learn:
• • • • What a polynomial is Add/subtract/multiply/divide polynomials Simplify radicals, exponents Solving equations with exponents and radicals • Complex numbers • Conjugates
POWER OF PRODUCT RULE
(
4 2x )⁵
x 2 x 32
5 4 5
20
( ab) ab
mn
n mn
m n
HAVE TO HAVE THE SAME BASE
QUOTIENT OF POWER RULE
3 3
10 4
3
6
x a b x b x
a
HAVE TO HAVE THE SAME BASE
POWER OF POWER RULE
(
4 x )³
x
m n
43
x
mn
12
(a ) a