节: 简化激进言论 | 11.4 简化激进言论

章节大纲

Simplification of Radical Expressions
::简化激进言论

When you add and subtract radical expressions, combine radical terms only when they have the same expression under the radical sign. This is a lot like combining like terms in variable expressions.
::当您添加和减去激进表达式时,只有在激进符号下有相同的表达式时,才会将激进术语组合起来。这非常像在变量表达式中将类似术语组合在一起。

Simplifying Expressions
::简化表达式

1. Simplify the following expressions as much as possible.
::1. 尽可能简化以下表达式。

a.) $\begin{aligned} 4 \sqrt{2} + 5 \sqrt{2} \end{aligned}$
::a. 42+52

$\begin{aligned} 4 \sqrt{2} + 5 \sqrt{2} = 9 \sqrt{2} \end{aligned}$

b.) $\begin{aligned} 2 \sqrt{3} - \sqrt{2} + 5 \sqrt{3} + 10 \sqrt{2} \end{aligned}$
::b) 23-2+53+102

$\begin{aligned} 2 \sqrt{3} - \sqrt{2} + 5 \sqrt{3} + 10 \sqrt{2} = 7 \sqrt{3} + 9 \sqrt{2} \end{aligned}$

It’s important to reduce all radicals to their simplest form in order to make sure that you ’re combining all possible like terms in the expression. For example, the expression $\begin{aligned} \sqrt{8} - 2 \sqrt{50} \end{aligned}$ looks like it can’t be simplified any more because it has no like terms. However, when you write each radical in its simplest form, you get $\begin{aligned} 2 \sqrt{2} - 10 \sqrt{2} \end{aligned}$ , and can combine those terms to get $\begin{aligned} - 8 \sqrt{2} \end{aligned}$ .
::重要的是将所有激进分子降低到最简单的形式,以确保你将所有可能的表达方式合并在一起。比如,8-250表达式似乎不能再简化了,因为它没有相似的表达式。然而,当你以最简单的形式写出每个激进分子时,你就会得到22-102,并且可以把这些词合并起来,以获得82。

2. Simplify the following expressions as much as possible.
::2. 尽可能简化以下表述。

a) $\begin{aligned} 4 \sqrt[3]{128} - \sqrt[3]{250} \end{aligned}$
:a) 41283-2503

$\begin{aligned} Re-write radicals in simplest terms: & = 4 \sqrt[3]{2 \cdot 64} - \sqrt[3]{2 \cdot 125} = 16 \sqrt[3]{2} - 5 \sqrt[3]{2} \\ Combine like terms: & = 11 \sqrt[3]{2} \end{aligned}$

::以最简单的术语重写基:=42643-21253=1623-523Combine 类似术语:=1123

b) $\begin{aligned} 3 \sqrt{x^{3}} - 4 x \sqrt{9 x} \end{aligned}$
:b) 3x3-4x9x

$\begin{aligned} Re-write radicals in simplest terms: & 3 \sqrt{x^{2} \cdot x} - 12 x \sqrt{x} & = 3 x \sqrt{x} - 12 x \sqrt{x} \\ Combine like terms: & = - 9 x \sqrt{x} \end{aligned}$

::以最简单的术语重写基:3x2xx-12xxx=3xx-12xxxCombine like terms:%9xx

Multiply Radical Expressions
::乘数激进表达式

When you multiply radical expressions, use the “raising a product to a power” rule: $\begin{aligned} \sqrt[m]{x \cdot y} = \sqrt[m]{x} \cdot \sqrt[m]{y} \end{aligned}$ . In the case of the next example, apply this rule in reverse.
::当您乘以激进表达式时,使用“将产品引向权力”规则: xym=xmym。在下一个示例中,将这一规则反向适用。

Simplify the expression $\begin{aligned} \sqrt{6} \cdot \sqrt{8} . \end{aligned}$
::简化表达式 68 。

$\begin{aligned} \sqrt{6} \cdot \sqrt{8} = \sqrt{6 \cdot 8} = \sqrt{48} \end{aligned}$

Or, in simplest radical form: $\begin{aligned} \sqrt{48} = \sqrt{16 \cdot 3} = 4 \sqrt{3} . \end{aligned}$
::或以最简单的激进形式: 48=16=3=43。

Another important fact is that $\begin{aligned} \sqrt{a} \cdot \sqrt{a} = \sqrt{a^{2}} = a . \end{aligned}$
::另一个重要事实是,Aa=a2=a。

When you multiply expressions that have numbers on both the outside and inside the radical sign, treat the numbers outside the radical sign and the numbers inside the radical sign separately. For example, $\begin{aligned} a \sqrt{b} \cdot c \sqrt{d} = a c \sqrt{b d} \end{aligned}$ .
::当您将带有激进符号内外数字的表达式乘以时,请将激进符号以外的数字和激进符号内的数字分开处理。例如 abcd=acbd。

Multiplying Expressions
::乘数表达式

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

U se distribution to eliminate the parentheses in each case.
::使用分布法去掉每种情况下的括号。

a) $\begin{aligned} \sqrt{2} (\sqrt{3} + \sqrt{5}) \end{aligned}$
:a) 2(3+5)

$\begin{aligned} Distribute \sqrt{2} inside the parentheses: & \sqrt{2} (\sqrt{3} + \sqrt{5}) & = \sqrt{2} \cdot \sqrt{3} + \sqrt{2} \cdot \sqrt{5} \\ Use the "raising a product to a power" rule: & = \sqrt{2 \cdot 3} + \sqrt{2 \cdot 5} \\ Simplify: & = \sqrt{6} + \sqrt{10} \end{aligned}$

::括号内分配 2 : (3+5) = 2}3+5 = 3+2 +2 5 使用“将产品增殖成动力” 规则:= 2=3+2 =5 简化:=6+10

b) $\begin{aligned} 2 \sqrt{x} (3 \sqrt{y} - \sqrt{x}) \end{aligned}$
:b) 2x(3y-x)

$\begin{aligned} Distribute 2 \sqrt{x} inside the parentheses: & = (2 \cdot 3) (\sqrt{x} \cdot \sqrt{y}) - 2 \cdot (\sqrt{x} \cdot \sqrt{x}) \\ Multiply: & = 6 \sqrt{x y} - 2 \sqrt{x^{2}} \\ Simplify: & = 6 \sqrt{x y} - 2 x \end{aligned}$

::括号内分布 2x : = (2)3 (xy)- 2(xx)- 2(xx) ltiply: = 6xy- 2x2 简化 := 6xy- 2xx

c) $\begin{aligned} (2 + \sqrt{5}) (2 - \sqrt{6}) \end{aligned}$
:c) (2+5)(2-6)

$\begin{aligned} Distribute: & (2 + \sqrt{5}) (2 - \sqrt{6}) & = (2 \cdot 2) - (2 \cdot \sqrt{6}) + (2 \cdot \sqrt{5}) - (\sqrt{5} \cdot \sqrt{6}) \\ Simplify: & = 4 - 2 \sqrt{6} + 2 \sqrt{5} - \sqrt{30} \end{aligned}$

::分发( 2+5)( 2 - 6) = ( 22) - ( 26) + ( 25) - ( 56) 简化: =4 - 26+25- 30

d) $\begin{aligned} (2 \sqrt{x} + 1) (5 - \sqrt{x}) \end{aligned}$
:d) (2x+1)(5-x)

$\begin{aligned} Distribute: & (2 \sqrt{x} - 1) (5 - \sqrt{x}) & = 10 \sqrt{x} - 2 x - 5 + \sqrt{x} \\ Simplify: & = 11 \sqrt{x} - 2 x - 5 \end{aligned}$

::分布 : (2x- 1) (5- x) = 10x-2x-5+x简单化 := 11x- 2x- 5

Rationalize the Denominator
::合理解析符号

Often, when you work with radicals, you end up with a radical expression in the denominator of a fraction. It’s traditional to write fractions in a form that doesn’t have radicals in the denominator, so you use a process called rationalizing the denominator to eliminate them.
::通常,当与激进分子合作时,你最终会在分母的分母中表现出激进的表达方式。传统上,以分母中没有激进分子的形式写分母,所以你用一个叫做理顺分母的过程来消除分母。

Rationalizing is easiest when there’s just a radical and nothing else in the denominator, as in the fraction $\begin{aligned} \frac{2}{\sqrt{3}} \end{aligned}$ . All you have to do then is multiply the numerator and denominator by a radical expression that makes the expression inside the radical into a perfect square, cube, or whatever power is appropriate. In the example above, multiply by $\begin{aligned} \sqrt{3} \end{aligned}$ :
::理性化是最容易的,当分母中只有一个激进的,没有其它的分母,就像第23项那样。然后,你要做的就是将分子和分母乘以一个激进表达式,使激进的表达方式变成一个完美的正方形、立方体或任何适当的力量。在以上的例子中,乘以3:

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

Cube roots and higher are a little trickier than square roots.
::立方根和高方根比平方根更狡猾。

Rationalizing Expressions
::合理化表达式

Rationalize $\begin{aligned} \frac{7}{\sqrt[3]{5}} \end{aligned}$
::合理化753

You can’t just multiply by $\begin{aligned} \sqrt[3]{5} \end{aligned}$ , because then the denominator would be $\begin{aligned} \sqrt[3]{5^{2}} \end{aligned}$ . To make the denominator a whole number, you need to multiply the numerator and the denominator by $\begin{aligned} \sqrt[3]{5^{2}} \end{aligned}$ :
::您不能仅仅乘以53, 因为分母就是523。要将分母变成一个整数, 您需要乘以分子和分母523:

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

Trickier still is when the expression in the denominator contains more than one term.
::当分母中的表达形式包含一个以上的术语时,就仍然会有特质。

Rationalizing Multi-Term Denominators
::合理解释多任期确定者

Consider the expression $\begin{aligned} \frac{2}{2 + \sqrt{3}} \end{aligned}$ . You can’t just multiply by $\begin{aligned} \sqrt{3} \end{aligned}$ , because you’d have to distribute that term and then the denominator would be $\begin{aligned} 2 \sqrt{3} + 3 \end{aligned}$ .
::考虑22+3的表达式。你不能仅仅乘以3, 因为您需要分配该词, 然后分母为 23+3 。

Instead, multiply by $\begin{aligned} 2 - \sqrt{3} \end{aligned}$ . This is a good choice because the product $\begin{aligned} (2 + \sqrt{3}) (2 - \sqrt{3}) \end{aligned}$ is a product of a sum and a difference , which means it’s a difference of squares. The radicals cancel each other out when you distribute , and the denominator works out to a single number !
$\begin{aligned} (2 + \sqrt{3}) (2 - \sqrt{3}) = 2^{2} - {(\sqrt{3})}^{2} = 4 - 3 = 1 \end{aligned}$

::相反,乘以 2-3 乘以 2-3 。这是一个很好的选择, 因为产品(2+3 (2-3) 是一个总和和差异的产物, 这意味着它是方形差异的产物。当分配时, 激进分子互相取消, 分母结果为单个数字 ! (2+3) (2-3) =22- (3)2=4-3=1

When you multiply both the numerator and denominator by $\begin{aligned} 2 - \sqrt{3} \end{aligned}$ , you end up with :
::当将分子和分母乘以 2 - 3 时, 最终结果为:

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

Now consider the expression $\begin{aligned} \frac{\sqrt{x} - 1}{\sqrt{x} - 2 \sqrt{y}} \end{aligned}$ .
::现在考虑 x- 1x-2y 表达式。

In order to eliminate the radical expressions in the denominator , multiply by $\begin{aligned} \sqrt{x} + 2 \sqrt{y} : \end{aligned}$
::为了消除分母中的激进表达式,乘以 x+2y:

Examples
::实例

Simplify the following expressions as much as possible.
::尽可能简化以下表达式。

Example 1
::例1

$\begin{aligned} 4 \sqrt{3} + 2 \sqrt{12} \end{aligned}$

$\begin{aligned} Simplify \sqrt{12} & = 4 \sqrt{3} + 2 \sqrt{4 \cdot 3} = 4 \sqrt{3} + 4 \sqrt{3} \\ Combine like terms: & = 8 \sqrt{3} \end{aligned}$

::简化 12 = 43+24}3 = 43 + 43 Combine 类似条件 := 83

Example 2
::例2

$\begin{aligned} 10 \sqrt{24} - \sqrt{28} \end{aligned}$

$\begin{aligned} Simplify \sqrt{24} and \sqrt{28} & = 10 \sqrt{6 \cdot 4} - \sqrt{7 \cdot 4} = 20 \sqrt{6} - 2 \sqrt{7} \\ There are no like terms. \end{aligned}$

::简化 24 和 28 = 1064 - 74= 206 - 27 没有类似的术语。

Review
::回顾

Simplify the following expressions as much as possible.
::尽可能简化以下表达式。

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

$\begin{aligned} \sqrt{180} + \sqrt{405} \end{aligned}$

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

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

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

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

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

$\begin{aligned} \frac{7}{\sqrt{5}} \end{aligned}$

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

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

$\begin{aligned} \frac{\sqrt{5}}{\sqrt{3 y}} \end{aligned}$
::53y 53y

$\begin{aligned} \frac{12}{2 - \sqrt{5}} \end{aligned}$

$\begin{aligned} \frac{6 + \sqrt{3}}{4 - \sqrt{3}} \end{aligned}$

$\begin{aligned} \frac{\sqrt{x}}{\sqrt{2} + \sqrt{x}} \end{aligned}$
::x2+xxx x2+xx

$\begin{aligned} \frac{5 y}{2 \sqrt{y} - 5} \end{aligned}$
::5y2y-5

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

Simplification of Radical Expressions ::简化激进言论

Simplifying Expressions ::简化表达式

Multiplying Expressions ::乘数表达式

Rationalize the Denominator ::合理解析符号

Rationalizing Expressions ::合理化表达式

Rationalizing Multi-Term Denominators ::合理解释多任期确定者

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)

Simplification of Radical Expressions
::简化激进言论

Simplifying Expressions
::简化表达式

Multiplying Expressions
::乘数表达式

Rationalize the Denominator
::合理解析符号

Rationalizing Expressions
::合理化表达式

Rationalizing Multi-Term Denominators
::合理解释多任期确定者

Examples
::实例

Example 1
::例1

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)