Section: 激进 | 1.5 激进派

Section outline

Introduction
::导言

What if you knew the area of a square was 1,000 square meters, and you wanted to find the length of its side? After completing this concept, you'll be able to find square roots like this one by hand and with a calculator.
::如果你知道一个广场的面积是1000平方米,你想找到它的侧边长度呢?完成这个概念之后,你就可以用手和计算器找到像这个一样的平方根。

Roots and Radicals
::根和激进

Square Roots
::平方根

A square root of a number is a number that, when multiplied by itself, gives the original number. In other words, if $\begin{aligned} a = b^{2} \end{aligned}$ , we say that $\begin{aligned} b \end{aligned}$ is the square root of $\begin{aligned} a \end{aligned}$ .
::数字的平方根是一个数字,当数字本身乘以时,就会给出原始数字。换句话说,如果a=b2,我们说b是a的平方根。

Note: Negative numbers and positive numbers both yield positive numbers when squared, so each positive number has both a positive and a negative square root. (For example, 3 and -3 can both be squared to yield 9.) The positive square root of a number is called the principal square root .
::注:负数和正数在平方时均产生正数,因此每个正数都有正数和负平方根。 (例如,3和-3均可平方得出9。 )一个数字的正平方根被称为主要平方根。

The square root of a number $\begin{aligned} x \end{aligned}$ is written as $\begin{aligned} \sqrt{x} \end{aligned}$ or sometimes as $\begin{aligned} \sqrt[2]{x} \end{aligned}$ . The symbol $\begin{aligned} \sqrt{} \end{aligned}$ is called a radical sign.
::数字 x 的平方根以 x 写成,有时以 x2 写成。符号称为激进符号。

Numbers with whole-number square roots are called perfect squares . The 1st five perfect squares (1, 4, 9, 16, and 25) are shown below.
::有整数平方根的数字称为完美的平方。下面列出前五个完美的平方(1、4、9、16和25)。

You can determine whether a number is a perfect square by looking at its prime factors. If every number in the factor tree appears an even number of times, the number is a perfect square. To find the square root of that number, simply take one of each pair of matching factors and multiply them together.
::您可以通过查看一个数字的质因数来确定数字是否为完美的正方形。如果系数树中的每个数字出现偶数, 数字是一个正方形。要找到该数字的平方根, 只需选择每对匹配系数中的一对, 然后将之相乘。

Simplifying Radical Expressions
::简化激进表达式

In general, t he $\begin{aligned} n \end{aligned}$ th root of a number $\begin{aligned} x \end{aligned}$ is written as $\begin{aligned} \sqrt[n]{x} \end{aligned}$ . The $\begin{aligned} n \end{aligned}$ is called the index , $\begin{aligned} \sqrt{} \end{aligned}$ is called the radical sign , and $\begin{aligned} x \end{aligned}$ is called the radicand . In the example $\begin{aligned} \sqrt[3]{5} \end{aligned}$ , "3" is the index and "5" is the radicand.
::一般而言,数字 x 的 nth 根写成 xn 。 n 称为索引,称为激进符号,x 称为弧形符号。在例53 中,“ 3” 表示索引,“ 5” 表示弧形符号。

The video below provides an overview of how to simplify radicals that are not perfect roots. The narrator provides learners with a four-step process that can be used to simplify radicals and models the process with examples.
::下面的视频概述了如何简化并非完美根基的激进分子。讲演者为学习者提供了一个四步过程,可以用来简化激进分子,并以实例来模拟这个过程。

Square Root Rules
::平根规则

Here are four rules that govern how we treat square roots:
::以下是四条规范我们如何对待平原的规则:

$\begin{aligned} \sqrt{a} \times \sqrt{b} = \sqrt{a b} \end{aligned}$
::axb=AB

$\begin{aligned} A \sqrt{a} \times B \sqrt{b} = A B \sqrt{a b} \end{aligned}$
::AaxBb=ABab

$\begin{aligned} \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \end{aligned}$ , $\begin{aligned} b \neq 0 \end{aligned}$
::ab=ab,b=0

$\begin{aligned} \frac{A \sqrt{a}}{B \sqrt{b}} = \frac{A}{B} \sqrt{\frac{a}{b}} \end{aligned}$ , $\begin{aligned} b \neq 0 \end{aligned}$
::AaBb=ABab,b0

Approximating Square Roots
::接近平方根

Terms like $\begin{aligned} \sqrt{2}, \sqrt{3} \end{aligned}$ , and $\begin{aligned} \sqrt{7} \end{aligned}$ (square roots of prime numbers) cannot be written as rational numbers . That is to say, they cannot be expressed as the ratio of two integers . We call them irrational numbers . In decimal form, an irrational number is a nonterminating and nonrepeating string of numbers after the decimal point.
::2,3和7等术语(质数的平方根)不能写成理性数字。也就是说,它们不能表示为两个整数之比。我们称之为非理性数字。以小数格式,非理性数字是小数点后不终止和不重复的数字字符串。

To find approximate values for square roots, we use the $\begin{aligned} \sqrt{} \end{aligned}$ or $\begin{aligned} \sqrt{x} \end{aligned}$ button on a calculator. When the number we plug in is a perfect square, or the square of a rational number , we will get an exact answer. When the number is a non-perfect square, the answer will be irrational and will look like a random string of digits. Since the calculator can only show some of the infinitely many digits that are actually in the answer, it is really showing us an approximate answer —not exactly the right answer, but as close as it can get.
::要找到正方根的近似值, 我们在计算器上使用或 x 按钮。当我们插入的数字是一个完美的正方或正方数时, 我们会得到一个准确的答案。当数字是一个不完美的正方形时, 答案将是不合理的, 并且看起来像一个随机的数字字符串。因为计算器只能显示答案中的一些无限多的数字, 它确实向我们展示了一个大致的答案, 不是准确的正确答案, 而是尽可能接近的答案。

Multiplying Radicals
::乘数激进

The following video provides an overview with examples of how to multiply and divide radicals:
::以下影片概述了如何增加和分裂激进分子的例子:

The rules for multiplication of square roots apply to $\begin{aligned} n \end{aligned}$ th roots provided they have the same index.
::平根乘法规则适用于nth根,只要它们具有相同的指数。

Multiplying Radicals
::乘数激进

$\begin{aligned} \sqrt[n]{a} \cdot \sqrt[n]{b} = \sqrt[n]{a b} where n is the index \end{aligned}$

Dividing Radicals
::分裂的激进派

Dividing radicals is more complicated. A radical in the denominator of a fraction is not considered simplified by mathematicians, because t raditional division requires dividing by an integer and not a irrational number. Since most radicals are irrational numbers, you must rationalize the denominator i n order to simplify the fraction.
::分裂的激进分子比较复杂。数学家并不认为分数分母中的激进分子可以简化,因为传统的分裂需要以整数而不是非理性数字来分隔。由于大多数激进分子都是非理性数字,你必须使分数合理化才能简化分数。

To rationalize the denominator means to remove any radical signs from the denominator of the fraction using multiplication.
::使分母合理化,即使用乘法清除分母分母的任何激进迹象。

Remember:
$\begin{aligned} \sqrt{a} \times \sqrt{a} = \sqrt{a^{2}} = a \end{aligned}$

::记住: axa=a2=a

Writing the Square Root as an Exponent
::将平方根写成指数

1. Evaluate $\begin{aligned} {(\sqrt{x})}^{2} \end{aligned}$ . What happens?
The $\begin{aligned} \sqrt{} \end{aligned}$ and the $\begin{aligned} ^{2} \end{aligned}$ cancel each other out, $\begin{aligned} ({\sqrt{x}}^{2}) = x \end{aligned}$ .
::1. 评价(x)2. 会发生什么?2和2相互取消,(x2)=x。

2. Recall that when a power is raised to another power, we multiply the exponents:
::2. 回顾当一个权力被提升到另一个权力时,我们乘以推手:

$\begin{aligned} {(\sqrt{x})}^{2} = {(x^{n})}^{2} = x^{n \cdot 2} = x^{1} = x \end{aligned}$

:x)2=(xn)2=(xn)2=(xn)2=(x)2=(x)1=(x)x

3. Thus, we can rewrite the exponents and root as an equation , $\begin{aligned} n \cdot 2 = 1 \end{aligned}$ . Solve for $\begin{aligned} n \end{aligned}$ :
$\begin{aligned} \frac{n \cdot 2}{2} & = \frac{1}{2} \\ n & = \frac{1}{2} \end{aligned}$

::3. 因此,我们可以重写前言和根方程,n2=1. 解决n:n22=12n=12

4. Therefore , $\begin{aligned} \sqrt{x} = x^{\frac{1}{2}} \end{aligned}$ .
::4. 因此,x=x12。

$\begin{aligned} {(\sqrt{x})}^{2} = {(x^{\frac{1}{2}})}^{2} = x^{(\frac{1}{2}) \cdot 2} = x^{1} = x \end{aligned}$

:x)2=(x12)2=x(12)2=x(x)2=x1=x

Similarly, evaluate $\begin{aligned} {(\sqrt[3]{x})}^{3} = x \end{aligned}$ .
::同样,评价(x3)3=x。

$\begin{aligned} {(\sqrt[3]{x})}^{3} = {(x^{\frac{1}{3}})}^{3} = x^{(\frac{1}{3}) \cdot 3} = x^{1} = x \end{aligned}$

:x3)3=(x13)3=(x13)3=(x13)3=(x13)3=x1=(x)

Thus, $\begin{aligned} \sqrt[3]{x} = x^{\frac{1}{3}} \end{aligned}$ .
::因此, x3=x13。

From this investigation, we can extend this idea to the other roots as well:
$\begin{aligned} \sqrt[4]{x} = x^{\frac{1}{4}}, \sqrt[5]{x} = x^{\frac{1}{5}}, \dots \sqrt[n]{x} = x^{\frac{1}{n}} . \end{aligned}$

::从这次调查中,我们可以把这个想法推广到其他根层: x4=x14, x5=x15,... xn=x1n。

The Rational Exponent Theorem :
::理性指數定理 :

For any real number $\begin{aligned} a \end{aligned}$ , index $\begin{aligned} n \end{aligned}$ , and exponent $\begin{aligned} m \end{aligned}$ , the following is always true:
::对于任何真实数字a、指数n和指数m,以下总是真实的:

$\begin{aligned} a^{\frac{m}{n}} = \sqrt[n]{a^{m}} = {(\sqrt[n]{a})}^{m} \end{aligned}$

::AMN=AMN=( an) m

Using a Calculator to Solve for Roots
::使用计算器解决根的计算器

To find $\begin{aligned} 5^{\frac{2}{3}} \end{aligned}$ using a calculator (if rounding your answer to the nearest hundredth).
::使用计算器(如果将答案四舍五入至最接近的百分百)找到523。

To type this into a calculator, the keystrokes would probably look like: $\begin{aligned} 5^(\frac{2}{3}) \end{aligned}$ .
::要将此项输入计算器, 键盘可能看起来像 : 5( 23) 。

The “^” symbol is used to indicate a power. Anything in parenthesis after the “^” would be in the exponent. Type: 5^(2/3). Evaluating this, the calculator would read 2.924017738... Round to the nearest hundredth for 2.92.
::“ ” 符号用于表示能量。 “ ” 之后的括号中的任何内容都会在引号中。类型 : 5( 2/3) 。评估这个符号时, 计算器将读为 2. 92 的 292401738 ... 圆到最接近的百分百。

Some calculators might have a $\begin{aligned} x^{y} \end{aligned}$ button. This button has the same purpose as the ^ and would be used in place of ^.
::某些计算器可能有一个 Xy 按钮。此按钮与 {} 具有相同的目的, 并且将用来代替 } 。

Adding and Subtracting Radicals
::增加和减减激进

Suppose you're taking a trip, and you'll be making two stops. This distance from your starting point to your 1st stop is $\begin{aligned} 14 \sqrt{2} \end{aligned}$ miles, and the distance from your 1st stop to your 2nd stop is $\begin{aligned} 9 \sqrt{2} \end{aligned}$ miles. How far will you travel in total? What operation would you have to perform to find the answer to this question? To add or subtract radicals, they must have the same root and radicand.
::假设你要去旅行,你会去两站。从起点到第一站的距离是142英里,从第一站到第二站的距离是92英里。你总共要走多远?你必须执行什么行动才能找到这个问题的答案?要增加或减去基数,它们必须有同样的根和线。

Adding and Subtracting Radicals
::增加和减减激进

$\begin{aligned} a \sqrt[n]{x} + b \sqrt[n]{x} = (a + b) \sqrt[n]{x} \end{aligned}$

The following video provides an overview of adding and subtracting radicals, including relevant vocabulary, steps, and examples:
::以下视频概述增减基数,包括相关词汇、步骤和实例:

Examples
::实例

Example 1
::例1

Determine the principal square root of each of these perfect squares:
::确定这些完美方块的主要平方根 :

a) 121
:a) 121

Solution:
::解决方案 :

$\begin{aligned} 121 = 11 \times 11 \end{aligned}$ , so $\begin{aligned} \sqrt{121} = 11 \end{aligned}$ .
::121=11x11, 所以121=11.

b) 225
::b) 225

Solution:
::解决方案 :

$\begin{aligned} 225 = (5 \times 5) \times (3 \times 3) \end{aligned}$ , so $\begin{aligned} \sqrt{225} = 5 \times 3 = 15 \end{aligned}$ .
::225=(5x5)x(3x3),225=5x3=15。

c) 324
:c) 324

Solution:
::解决方案 :

$\begin{aligned} 324 = (2 \times 2) \times (3 \times 3) \times (3 \times 3) \end{aligned}$ , so $\begin{aligned} \sqrt{324} = 2 \times 3 \times 3 = 18 \end{aligned}$ .
::324=(2x2)x(3x3)x(3x3)x(3x3),因此324=2x3x3=18。

When the prime factors don’t pair up neatly, we “factor out” the ones that do pair up and leave the rest under a radical sign. We write the answer as $\begin{aligned} a \sqrt{b} \end{aligned}$ , where $\begin{aligned} a \end{aligned}$ is the product of half the paired factors we pulled out and $\begin{aligned} b \end{aligned}$ is the product of the leftover factors.
::当主要因素不完美地对齐时,我们就“把配对的因素归结在一起 ” , 把其余的都留在一个激进的标志之下。我们把答案写成 ab, 其中一种是一半配对因素的产物,我们拿出来,b 是剩余因素的产物。

Example 2
::例2

Determine principal or positive square root of the following numbers:
::确定下列数字的主根或正平方根:

a) 8
:a) 8

Solution: $\begin{aligned} \sqrt{8} = \sqrt{(2 \times 2) \times 2} = 2 \sqrt{2} \end{aligned}$ . This gives us one pair of 2s and one leftover 2 inside the square root.
::解析度: 8=( 2x2) xx2=22。这样一对 2 和 2 在平方根内留一个 2 。

b) 48
:b) 48

Solution:
::解决方案 :

$\begin{aligned} \sqrt{48} = \sqrt{(2 \times 2 \times 2 \times 2) \times 3} = 4 \sqrt{3} \end{aligned}$

c) 75
:c) 75

Solution:
::解决方案 :

$\begin{aligned} \sqrt{75} = \sqrt{(5 \times 5) \times 3} = 5 \sqrt{3} \end{aligned}$

Example 3
::例3

Simplify the following square-root problems:
::简化以下平方根问题:

a) $\begin{aligned} \sqrt{8} \times \sqrt{2} \end{aligned}$
::a) 8x2

Solution:
::解决方案 :

$\begin{aligned} \sqrt{8} \times \sqrt{2} = \sqrt{16} = 4 \end{aligned}$

b) $\begin{aligned} 3 \sqrt{4} \times 4 \sqrt{3} \end{aligned}$
:b) 34×43

Solution:
::解决方案 :

$\begin{aligned} 3 \sqrt{4} \times 4 \sqrt{3} = 12 \sqrt{12} = 12 \sqrt{(2 \times 2) \times 3} = 12 \times 2 \sqrt{3} = 24 \sqrt{3} \end{aligned}$

c) $\begin{aligned} \sqrt{12} \div \sqrt{3} \end{aligned}$
:c) 12+3

Solution:
::解决方案 :

$\begin{aligned} \sqrt{12} \div \sqrt{3} = \sqrt{\frac{12}{3}} = \sqrt{4} = 2 \end{aligned}$

d) $\begin{aligned} 12 \sqrt{10} \div 6 \sqrt{5} \end{aligned}$
:d) 121065

Solution:
::解决方案 :

$\begin{aligned} 12 \sqrt{10} \div 6 \sqrt{5} = \frac{12}{6} \sqrt{\frac{10}{5}} = 2 \sqrt{2} \end{aligned}$

Example 4
::例4

Use a calculator to find the following square roots. Round your answer to three decimal places.
::使用计算器查找以下的正方根。将您的答复四舍五入到小数点后三位。

a) $\begin{aligned} \sqrt{99} \end{aligned}$
:a) 99

Solution:
::解决方案 :

$\begin{aligned} \approx 9.950 \end{aligned}$

b) $\begin{aligned} \sqrt{0.5} \end{aligned}$
::b) 0.5

Solution:
::解决方案 :

$\begin{aligned} \approx 0.707 \end{aligned}$

c) $\begin{aligned} \sqrt{1.75} \end{aligned}$
:c) 1.75

Solution:
::解决方案 :

$\begin{aligned} \approx 1.323 \end{aligned}$

d) $\begin{aligned} \sqrt[7]{12} \end{aligned}$
:d) 127

Solution:
::解决方案 :

Using rational exponents, the $\begin{aligned} 7^{t h} \end{aligned}$ root becomes the $\begin{aligned} \frac{1}{7} \end{aligned}$ power; $\begin{aligned} 12^{\frac{1}{7}} \approx 1.426 \end{aligned}$ .
::使用理性的推论, 第7根成为17个功率; 12171.426。

Example 5
::例5

Simplify $\begin{aligned} \sqrt{3} \cdot \sqrt{12} \end{aligned}$ .
::简化 312 。

Solution:
::解决方案 :

$\begin{aligned} \sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6 \end{aligned}$

Example 6
::例6

Simplify $\begin{aligned} \frac{2}{\sqrt{3}} \end{aligned}$ .
::简化 23

Solution:
::解决方案 :

We must clear the denominator of its radical using the property above. Remember, what you do to one piece of a fraction, you must do to all pieces of the fraction:
::我们必须用上面的属性来清除它激进的分母。记住, 你对其中的一小部分做了什么, 您必须对所有的分母做些什么 :

$\begin{aligned} \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2 \sqrt{3}}{\sqrt{3^{2}}} = \frac{2 \sqrt{3}}{3} \end{aligned}$

Let’s now consider how to convert roots into exponents. Look at the square root and see if we can use the properties of exponents to determine what exponential number it is equivalent to.
::现在让我们来考虑如何将根转换成指数。看看平方根,看看我们能否使用指数的特性来确定指数的指数值。

Example 7
::例7

Simplify the following radical expressions:
::简化以下激进表达式:

a) $\begin{aligned} 256^{\frac{1}{4}} \end{aligned}$
:a) 25614

Solution:
::解决方案 :

Rewrite this expression as a radical expression . A number to the one-fourth power is the same as the 4th root:
::重写此表达式为激进表达式。四分之一权力的数与 4 根相同 :

$\begin{aligned} 256^{\frac{1}{4}} = \sqrt[4]{256} = \sqrt[4]{4^{4}} = 4 \end{aligned}$

Therefore, $\begin{aligned} 256^{\frac{1}{4}} = 4 \end{aligned}$ .
::因此,25614=4。

b) $\begin{aligned} 49^{\frac{3}{2}} \end{aligned}$
:b) 4932

Solution:
::解决方案 :

In this problem, the root is written as an exponent. Rewrite the problem as a radical expression:
::将问题重写为激进的表达方式:

$\begin{aligned} 49^{\frac{3}{2}} = {(49^{3})}^{\frac{1}{2}} = \sqrt{49^{3}} \end{aligned}$ or $\begin{aligned} {(\sqrt{49})}^{3} \end{aligned}$
::4932=(493)12=493或(49)3

It may be easier to evaluate the 2nd option above: $\begin{aligned} {(\sqrt{49})}^{3} = 7^{3} = 343 \end{aligned}$ or $\begin{aligned} {(\sqrt{49})}^{3} = \sqrt{49} \cdot \sqrt{49} \cdot \sqrt{49} = 7 \cdot 7 \cdot 7 = 343 \end{aligned}$ .
::评估上述第2种选择可能比较容易493=73=343或(493=494949=77=343)。

c) $\begin{aligned} 125^{\frac{4}{3}} \end{aligned}$
:c) 12543

Solution:
::解决方案 :

$\begin{aligned} 125^{\frac{4}{3}} = {(\sqrt[3]{125})}^{4} = 5^{4} = 625 \end{aligned}$

d) $\begin{aligned} 256^{\frac{5}{8}} \end{aligned}$
:d) 25658

Solution:
::解决方案 :

$\begin{aligned} 256^{\frac{5}{8}} = {(\sqrt[8]{256})}^{5} = 2^{5} = 32 \end{aligned}$

Example 8
::例8

Simplify $\begin{aligned} \frac{7}{\sqrt[3]{5}} \end{aligned}$ .
::简化 753 。

Solution:
::解决方案 :

In this case, we need to make the number inside the cube root a perfect cube. We need to multiply the numerator and the denominator by $\begin{aligned} \sqrt[3]{5^{2}} \end{aligned}$ .
::在此情况下, 我们需要将立方根内的数字做成一个完美的立方体。我们需要将分子和分母乘以 523 。

$\begin{aligned} \frac{7}{\sqrt[3]{5}} \cdot \frac{\sqrt[3]{5^{2}}}{\sqrt[3]{5^{2}}} = \frac{7 \sqrt[3]{25}}{\sqrt[3]{5^{3}}} = \frac{7 \sqrt[3]{25}}{5} \end{aligned}$

Example 9

::例9

A planet's maximum distance from the sun (in astronomical units) is given by the formula $\begin{aligned} d = p^{\frac{2}{3}} \end{aligned}$ where p is the period (in years) of the planet's orbit around the sun. If a planet's orbit around the sun is 27 years, what is its distance from the sun?
::公式d=p23给出了行星与太阳的最大距离(天文单位 ) 。公式d=p23给出了行星绕太阳轨道运行的时间( 年) 。如果行星绕太阳运行的轨道是27年,那么它与太阳的距离是多少?

Solution:
::解决方案 :

Substitute 27 for p and solve.
::代替27号的p和p和p的解答。

$\begin{aligned} d = 27^{\frac{2}{3}} \end{aligned}$
::d=2723 (d=2723)

Rewrite the problem:
::重写问题 :

$\begin{aligned} 27^{\frac{2}{3}} = {(27^{2})}^{\frac{1}{3}} = \sqrt[3]{27^{2}} \end{aligned}$ or $\begin{aligned} {\sqrt[3]{27}}^{2} \end{aligned}$
::2723=(272)13=2723或2732

$\begin{aligned} {(\sqrt[3]{27})}^{2} = 3^{2} = 9 \end{aligned}$ .

Therefore, the planet's distance from the sun is 9 astronomical units.
::因此,地球与太阳的距离是9个天文单位。

Example 10
::例10

Simplify the following radical expressions:
::简化以下激进表达式:

a) $\begin{aligned} 3 \sqrt{5} + 6 \sqrt{5} \end{aligned}$ .
:a) 35+65。

Solution:
::解决方案 :

Since both terms in the sum have a factor of “ $\begin{aligned} \sqrt{5} \end{aligned}$ ”, this factor is treated similar to like terms . Using the rule above:
::由于总和中的这两个词都有“5”因数,这一因数与类似的因数处理类似。

$\begin{aligned} 3 \sqrt{5} + 6 \sqrt{5} = (3 + 6) \sqrt{5} = 9 \sqrt{5} \end{aligned}$

b) $\begin{aligned} 2 \sqrt[3]{13} + 6 \sqrt[3]{12} \end{aligned}$ .
:b) 2133+6123。

Solution:
::解决方案 :

Since the two terms do have have the same radical factor, there can be no further simplification.
::由于这两个词确实有着相同的根本因素,因此无法进一步简化。

In some cases, the radical may need to be reduced before addition/ subtraction is possible.
::在某些情况下,在可能增加/减量之前,可能需要减少激进性。

c) $\begin{aligned} 4 \sqrt{3} + 2 \sqrt{12} \end{aligned}$ .
:c) 43+212。

Solution:
::解决方案 :

The first term is already simplified and $\begin{aligned} \sqrt{12} \end{aligned}$ simplifies to $\begin{aligned} 2 \sqrt{3} \end{aligned}$ .
::第一个任期已经简化,12个任期简化为23个。

$\begin{aligned} 4 \sqrt{3} + 2 \sqrt{12} & = 4 \sqrt{3} + 2 (2 \sqrt{3}) \\ 4 \sqrt{3} + 4 \sqrt{3} & = 8 \sqrt{3} \end{aligned}$

d) $\begin{aligned} 3 \sqrt[3]{2} + 5 \sqrt[3]{16} \end{aligned}$ .
::d) 323+5163。

Solution:
::解决方案 :

Step 1: Factor the 2nd radical to simplify:
::第1步:将第2步基乘以简化:

$\begin{aligned} 3 \sqrt[3]{2} + 5 \sqrt[3]{16} & = 3 \sqrt[3]{2} + 5 \sqrt[3]{2 \cdot 8} \\ = 3 \sqrt[3]{2} + 5 \sqrt[3]{2 \cdot 2^{3}} \\ = 3 \sqrt[3]{2} + 5 \sqrt[3]{2^{3}} \cdot \sqrt[3]{2} \\ = 3 \sqrt[3]{2} + 5 \cdot 2 \sqrt[3]{2} \\ = 3 \sqrt[3]{2} + 10 \sqrt[3]{2} \end{aligned}$

Step 2: Combine the two radicals:
$\begin{aligned} 3 \sqrt[3]{2} + 10 \sqrt[3]{2} & = (3 + 10) \sqrt[3]{2} \\ = 13 \sqrt[3]{2} \end{aligned}$

::步骤2:将两根基合并:323+1023=(3+10)23=1323

Review
::回顾

For 1-10, find the following square roots exactly without using a calculator, giving your answer in the simplest form:
::1-10, 在不使用计算器的情况下, 找到以下的正方根, 以最简单的形式回答 :

$\begin{aligned} \sqrt{25} \end{aligned}$

$\begin{aligned} \sqrt{24} \end{aligned}$

$\begin{aligned} \sqrt{2000} \end{aligned}$

$\begin{aligned} \sqrt{\frac{1}{4}} \end{aligned}$ (Hint: The division rules you learned can be applied backwards!)
::14 (提示: 您所学的分区规则可以被反向应用 !)

$\begin{aligned} \sqrt{\frac{9}{4}} \end{aligned}$

$\begin{aligned} \sqrt{0.16} \end{aligned}$

$\begin{aligned} \sqrt{0.01} \end{aligned}$

For 8-15, use a calculator to solve. Round to two decimal places.
::8-15 使用计算器解析。小数点到小数点后两位位数。

$\begin{aligned} \sqrt{13} \end{aligned}$

$\begin{aligned} \sqrt{99} \end{aligned}$

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

$\begin{aligned} \sqrt{2000} \end{aligned}$

$\begin{aligned} \sqrt{.25} \end{aligned}$

$\begin{aligned} \sqrt{1.35} \end{aligned}$

$\begin{aligned} \sqrt{0.37} \end{aligned}$

$\begin{aligned} \sqrt{0.01} \end{aligned}$

Rewrite using rational exponents or roots, and use a calculator for the problems below. Round to two decimal places.
::使用合理的指数或根重写,并对下面的问题使用计算器。小数点到小数点后两舍五入。

$\begin{aligned} \sqrt[5]{45} \end{aligned}$

$\begin{aligned} \sqrt[9]{140} \end{aligned}$

$\begin{aligned} {\sqrt[8]{50}}^{3} \end{aligned}$

$\begin{aligned} 72^{\frac{5}{3}} \end{aligned}$

$\begin{aligned} 125^{\frac{3}{4}} \end{aligned}$

Multiply the following expressions:
::乘以下列表达式:

$\begin{aligned} \sqrt{6} (\sqrt{10} + \sqrt{8}) \end{aligned}$

$\begin{aligned} (\sqrt{a} - \sqrt{b}) (\sqrt{a} + \sqrt{b}) \end{aligned}$
:a-b)(a+b)

$\begin{aligned} (2 \sqrt{x} + 5) (2 \sqrt{x} + 5) \end{aligned}$
:2x+5)(2x+5)

Rationalize the denominator:
::理顺分母 :

$\begin{aligned} \frac{7}{\sqrt{15}} \end{aligned}$

$\begin{aligned} \frac{9}{\sqrt{10}} \end{aligned}$

$\begin{aligned} \frac{2 x}{\sqrt{5} x} \end{aligned}$
::2x5x

Evaluate the following without a calculator:
::在无计算器的情况下评价以下内容:

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

$\begin{aligned} 27^{\frac{4}{3}} \end{aligned}$

$\begin{aligned} 16^{\frac{5}{4}} \end{aligned}$

$\begin{aligned} \sqrt{25^{3}} \end{aligned}$

$\begin{aligned} {\sqrt[2]{9}}^{5} \end{aligned}$

$\begin{aligned} \sqrt[5]{32^{2}} \end{aligned}$

For the following problems, rewrite the expressions with rational exponents and then simplify the exponent and evaluate without a calculator:
::对于下列问题,请用理性的推算符重写表达式,然后简化推算符,不使用计算器进行评估:

$\begin{aligned} \sqrt[4]{{(\frac{2}{3})}^{8}} \end{aligned}$

$\begin{aligned} {\sqrt[3]{\frac{7}{2}}}^{6} \end{aligned}$

$\begin{aligned} {\sqrt{{(16)}^{\frac{1}{2}}}}^{6} \end{aligned}$

Write the following expressions in simplest radical form:
::以最简单的激进形式写下以下表达式:

$\begin{aligned} \sqrt[3]{48 a^{3} b^{7}} \end{aligned}$
::48a3b73

$\begin{aligned} \sqrt[3]{\frac{16 x^{5}}{135 y^{4}}} \end{aligned}$
::16x5135y43

True or false? $\begin{aligned} \sqrt[7]{5} \cdot \sqrt[6]{6} = \sqrt[42]{30} \end{aligned}$
::真实还是假? 57_66=3042

$\begin{aligned} 3 \sqrt{8} - 6 \sqrt{32} \end{aligned}$

$\begin{aligned} \sqrt{6} - \sqrt{27} + 2 \sqrt{54} + 3 \sqrt{48} \end{aligned}$

$\begin{aligned} \sqrt{8 x^{3}} - 4 x \sqrt{98 x} \end{aligned}$
::8x3 - 4x98x

$\begin{aligned} \sqrt{48 a} + \sqrt{27 a} \end{aligned}$
::48a+27a

$\begin{aligned} \sqrt[3]{4 x^{3}} + x \sqrt[3]{256} \end{aligned}$
::433+2563号

Solve the following:
::解决以下问题:

The volume of a spherical balloon is $\begin{aligned} 950 c m^{3} \end{aligned}$ . Find the radius of the balloon. (Volume of a sphere $\begin{aligned} = \frac{4}{3} π R^{3} \end{aligned}$ )
::球球球气球的体积为950厘米3. 找到气球的半径。 (球体卷=431R3)

The volume of a soda can is $\begin{aligned} 355 c m^{3} \end{aligned}$ . The height of the can is 4 times the radius of the base. Find the radius of the base of the cylinder.
::苏打罐的体积是355厘米3,罐头的高度是基底半径的4倍。找到圆筒底部的半径。

Review (Answers)
::回顾(答复)

Please see the Appendix.
::请参看附录。

Section outline

Introduction ::导言

Roots and Radicals ::根和激进

Square Roots ::平方根

Simplifying Radical Expressions ::简化激进表达式

Square Root Rules ::平根规则

Approximating Square Roots ::接近平方根

Multiplying Radicals ::乘数激进

Multiplying Radicals ::乘数激进

Dividing Radicals ::分裂的激进派

Writing the Square Root as an Exponent ::将平方根写成指数

The Rational Exponent Theorem : ::理性指數定理 :

Using a Calculator to Solve for Roots ::使用计算器解决根的计算器

Adding and Subtracting Radicals ::增加和减减激进

Adding and Subtracting Radicals ::增加和减减激进

Examples ::实例

Example 10 ::例10

Review ::回顾

Review (Answers) ::回顾(答复)

Introduction
::导言

Roots and Radicals
::根和激进

Square Roots
::平方根

Simplifying Radical Expressions
::简化激进表达式

Square Root Rules
::平根规则

Approximating Square Roots
::接近平方根

Multiplying Radicals
::乘数激进

Multiplying Radicals
::乘数激进

Dividing Radicals
::分裂的激进派

Writing the Square Root as an Exponent
::将平方根写成指数

The Rational Exponent Theorem :
::理性指數定理 :

Using a Calculator to Solve for Roots
::使用计算器解决根的计算器

Adding and Subtracting Radicals
::增加和减减激进

Adding and Subtracting Radicals
::增加和减减激进

Examples
::实例

Example 10
::例10

Review
::回顾

Review (Answers)
::回顾(答复)