11.3 提高产品或电力的报价

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  提高产品或对电力的报价

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  Raising a Product or a Quotient to a Power
  ::提高产品或电力报价

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  A radical reverses the operation of raising a number to a power. For example, the square of 4 is $\begin{align*}4^2 = 4 \cdot 4 = 16\end{align*}$ , and so the square root of 16 is 4. The symbol for a square root is $\begin{align*}\sqrt{\;\;}\end{align*}$ . This symbol is also called the radical sign.
  ::激进地颠倒将数数增长到权力的操作。 例如, 4的平方是42=44=16, 16的平方根是4。 平方根的符号是。 这个符号也被称为激进符号 。

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  In addition to square roots, we can also take cube roots, fourth roots, and so on. For example, since 64 is the cube of 4, 4 is the cube root of 64.
  ::除了平方根之外,我们还可以选择立方根、四根根等等。例如,64是4的立方根,4是64的立方根。

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  $$\begin{align*}\sqrt[3]{64} = 4 \qquad \text{since} \qquad 4^3 = 4 \cdot 4 \cdot 4 = 64\end{align*}$$
  ::364=4自43=444=64

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  We put an index number in the top left corner of the radical sign to show which root of the number we are seeking. Square roots have an index of 2, but we usually don’t bother to write that out.
  ::我们在激进标志左上角放置了一个索引号,以显示我们所寻求的数字的根。 方根指数为2, 但我们通常不费心写出来。

  $$\begin{align*}\sqrt[2]{36} = \sqrt{36} = 6\end{align*}$$

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  The cube root of a number gives a number which when raised to the power three gives the number under the radical sign. The fourth root of number gives a number which when raised to the power four gives the number under the radical sign:
  ::数字的立方根给出一个数字,当被加到三号电源时,该数字在激进符号下给出。数字的第四个根给出了一个数字,当加到四号电源时,该数字在激进符号下给出数字:

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  $$\begin{align*}\sqrt[4]{81} = 3 \qquad \text{since} \qquad 3^4 = 3 \cdot 3 \cdot 3 \cdot 3 = 81\end{align*}$$
  ::481=3自34=3333=81

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  And so on for any power we can name.
  ::任何能命名的能量,等等。

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  Even and Odd Roots
  ::平平的根,

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  Radical expressions that have even indices are called even roots and radical expressions that have odd indices are called odd roots . There is a very important difference between even and odd roots, because they give drastically different results when the number inside the radical sign is negative.
  ::甚至具有指数的激进表达方式甚至被称为根,而带有奇数的激进表达方式被称为奇数根。 偶数和奇数根之间有着非常重要的区别,因为当激进符号内的数字为负数时,它们会产生截然不同的结果。

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  Any real number raised to an even power results in a positive answer. Therefore , when the index of a radical is even, the number inside the radical sign must be non-negative in order to get a real answer.
  ::任何真正的数字被提升到一个平权,都会得到肯定的答案。 因此,当激进指数达到平衡时,激进标志内的数字必须是非负的,才能得到真正的答案。

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  On the other hand, a positive number raised to an odd power is positive and a negative number raised to an odd power is negative. Thus, a negative number inside the radical sign is not a problem. It just results in a negative answer.
  ::另一方面,向奇异力量提出的正数是正数,而向奇异力量提出的负数是负数。 因此,激进迹象中的负数不是问题。 它只会导致否定回答。

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  Evaluating Radical Expressions 
  ::评价激进言论

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

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  a)  $\begin{align*}\sqrt{121}\end{align*}$
  :a) 121

  $\begin{align*}\sqrt{121} = 11\end{align*}$

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  b)  $\begin{align*}\sqrt[3]{125}\end{align*}$
  :b) 3125

  $\begin{align*}\sqrt[3]{125} = 5\end{align*}$

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  c)  $\begin{align*}\sqrt[4]{-625}\end{align*}$
  :c) 4625

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  $\begin{align*}\sqrt[4]{-625}\end{align*}$ is not a real number
  ::4625不是真实数字

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  d)  $\begin{align*}\sqrt[5]{-32}\end{align*}$
  ::d) 532

  $\begin{align*}\sqrt[5]{-32} = -2\end{align*}$

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  Using the Product and Quotient Properties of Radicals
  ::使用激进分子的产品和引号属性

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  Radicals can be re-written as rational powers. The radical: $\begin{align*}\sqrt[m]{a^n}\end{align*}$ is defined as $\begin{align*}a^{\frac{n}{m}}\end{align*}$ .
  ::激进可以被改写为理性力量。 激进: man 被定义为 am 。

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  Write each expression as an exponent with a rational value for the exponent.
  ::将每个表达式写成一个引言, 给引言者提供一个合理的值 。

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  a) $\begin{align*}\sqrt{5}\end{align*}$
  :a) 5

  $\begin{align*}\sqrt{5} = 5^{\frac{1}{2}}\end{align*}$

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  b) $\begin{align*}\sqrt[4]{a}\end{align*}$
  :b) 4a

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  $\begin{align*}\sqrt[4]{a} = a^{\frac{1}{4}}\end{align*}$
  ::4a=a14

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  c) $\begin{align*} \sqrt[3]{4xy}\end{align*}$
  :c) 34xy

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  $\begin{align*} \sqrt[3]{4xy} = (4xy)^{\frac{1}{3}}\end{align*}$
  ::34xy=( 4xy)13

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  d) $\begin{align*}\sqrt[6]{x^5}\end{align*}$
  :d) 6xx5

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  $\begin{align*}\sqrt[6]{x^5} = x^{\frac{5}{6}}\end{align*}$
  ::6x5=x56

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  As a result of this property , for any non-negative number $\begin{align*}a\end{align*}$ we know that $\begin{align*}\sqrt[n]{a^n} = a^{\frac{n}{n}} = a\end{align*}$ .
  ::由于这种财产,对于任何非负数,我们知道 nan=ann=a。

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  Since roots of numbers can be treated as powers, we can use exponent rules to simplify and evaluate radical expressions. Let’s review the product and quotient rule of exponents.
  ::由于数字根源可以被视为权力,我们可以使用推论规则来简化和评估激进表达方式。 让我们来审查产品和推论者商数规则。

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  $$\begin{align*}\text{Raising a product to a power:} && (x \cdot y)^n & = x^n \cdot y^n\\ \text{Raising a quotient to a power:} && \left(\frac{x}{y} \right)^n & = \frac{x^n}{y^n}\end{align*}$$
  ::将产品提升到电源 : (xy) n=xnynyn 将商数提升到电源 : (xy) n=xnyn

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  In radical notation, these properties are written as
  ::在粗略的标记中,这些财产的写法是:

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  $$\begin{align*}\text{Raising a product to a power:} && \sqrt[m]{x \cdot y} & = \sqrt[m]{x} \ \cdot \sqrt[m]{y}\\ \text{Raising a quotient to a power:} && \sqrt[m]{\frac{x}{y}} & = \frac{\sqrt[m]{x}}{\sqrt[m]{y}}\end{align*}$$
  ::将产品提升到电源:mxy=mxxy 将商数提升到电源:mxy=mxy

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  A very important application of these rules is reducing a radical expression to its simplest form. This means that we apply the root on all the factors of the number that are perfect roots and leave all factors that are not perfect roots inside the radical sign.
  ::这些规则的一个重要应用方式是将激进的表达方式降低到最简单的形式。 这意味着我们把根本根植于数字的所有因素上,将所有不完全根植于激进标志中的因素都置于根本基础之上。

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  For example, in the expression $\begin{align*}\sqrt{16}\end{align*}$ , the number 16 is a perfect square because $\begin{align*}16 = 4^2\end{align*}$ . This means that we can simplify it as follows:
  ::例如,在"% 16"的表达式中,数字16是一个完美的方形,因为16=42。这意味着我们可以简化如下:

  $$\begin{align*}\sqrt{16} = \sqrt{4^2} = 4\end{align*}$$

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  Thus, the square root disappears completely.
  ::因此,平方根完全消失。

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  On the other hand, in the expression $\begin{align*}\sqrt{32}\end{align*}$ , the number 32 is not a perfect square, so we can’t just remove the square root. However, we notice that $\begin{align*}32 = 16 \cdot 2\end{align*}$ , so we can write 32 as the product of a perfect square and another number. Thus,
  ::另一方面,在“三十二”的表达方式中,数字32不是一个完美的正方形,因此我们不能直接去除平方根。 然而,我们注意到32=162,因此我们可以将32写成一个完美的正方形和另一个数字。 因此,

  $$\begin{align*}\sqrt{32} = \sqrt{16 \cdot 2}\end{align*}$$

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  If we apply the “raising a product to a power” rule we get:
  ::如果我们将“将产品用于权力”规则,我们就会得到:

  $$\begin{align*}\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \ \cdot \sqrt{2}\end{align*}$$

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  Since $\begin{align*}\sqrt{16} = 4\end{align*}$ , we get: $\begin{align*}\sqrt{32} = 4 \cdot \sqrt{2} = \underline{\underline{4 \sqrt{2}}}\end{align*}$
  ::从16=4开始,我们得到:32=4=2=4

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  Writing Expressions in the Simplest Radical Form 
  ::简单激进表格中的写法表达式

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  1. Write the following expressions in the simplest radical form.
  ::1. 以最简单的激进形式写下以下表述。

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  The strategy is to write the number under the square root as the product of a perfect square and another number. The goal is to find the highest perfect square possible; if we don’t find it right away, we just repeat the procedure until we can’t simplify any longer.
  ::战略是将数字写在平方根下,作为完美的平方和另一个数字的产物。 目标是找到尽可能高的完全的平方;如果我们不马上找到,我们就重复程序,直到我们无法再简化。

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  a)  $\begin{align*}\sqrt{8}\end{align*}$
  :a) ____________________________________________________________________________________________________________________________________

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  $$\begin{align*}\text{We can write} \ 8 = 4 \cdot 2, \ \text{so} \sqrt{8} = \sqrt{4 \cdot 2}.\\ \text{With the Raising a product to a power rule, that becomes} \sqrt{4} \ \cdot \sqrt{2}.\\ \text{Evaluate} \ \sqrt{4} \ \text{and we're left with } \underline{\underline{2\sqrt{2}}}.\end{align*}$$
  ::我们可以写8=42, 所以842. 随着产品 提升到权力规则, 成为42. Evaluate_4,我们只剩下22。

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  b)  $\begin{align*} \sqrt{50}\end{align*}$
  ::b) ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  $$\begin{align*}\text{We can write} \ 50 = 25 \cdot 2, \ \text{so:} \sqrt{50} = \sqrt{25 \cdot 2}\\ \text{Use the Raising a product to a power rule:} =\sqrt{25} \ \cdot \sqrt{2} = \underline{\underline{5 \sqrt{2}}}\end{align*}$$
  ::我们可以写50=252,所以:"50252" 将产品用于发电规则:"252=52"

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  c)  $\begin{align*}\sqrt{\frac{125}{72}}\end{align*}$
  ::c) 12572

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  $$\begin{align*}\text{Use the Raising a quotient to a power rule to separate the fraction:} \sqrt{\frac{125}{72}} = \frac{\sqrt{125}}{\sqrt{72}}\\ \text{Re-write each radical as a product of a perfect square and another number:} = \frac{ \sqrt{25 \cdot 5}}{\sqrt{36 \cdot 2}} = \frac{5 \sqrt{5}}{6 \sqrt{2}}\end{align*}$$
  ::用提高商数到权力规则来分离分数: 1257212572 re- write each rotic as a perfect squal and another number: 255362=5562。

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  The same method can be applied to reduce radicals of different indices to their simplest form.
  ::同一方法可用于将不同指数的基质减到最简单的形式。

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  2. Write the following expression in the simplest radical form.
  ::2. 以最简单的激进形式写下以下表述。

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  In these cases we look for the highest possible perfect cube, fourth power, etc. as indicated by the index of the radical.
  ::在这些情况下,我们寻找尽可能高的完美立方体,第四力量,等等,正如激进分子指数所显示的。

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  a)  $\begin{align*}\sqrt[3]{40}\end{align*}$
  :a) 3___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

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  Here we are looking for the product of the highest perfect cube and another number. We write: $\begin{align*}\sqrt[3]{40} = \sqrt[3]{8 \cdot 5} = \sqrt[3]{8} \ \cdot \sqrt[3]{5} = 2 \sqrt[3]{5}\end{align*}$
  ::我们在这里寻找最高完美立方体和另一个数字的产物。 我们写: 340= 385= 38= 35=235。

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  b)  $\begin{align*}\sqrt[4]{\frac{162}{80}}\end{align*}$
  :b) 416280

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  Here we are looking for the product of the highest perfect fourth power and another number.
  ::我们在这里寻找 最高完美的第四强的产物 和另一个数字。

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  $$\begin{align*}\text{Re-write as the quotient of two radicals:} && \sqrt[4]{\frac{162}{80}} & = \frac{\sqrt[4]{162}}{\sqrt[4]{80}}\\ \text{Simplify each radical separately:} && & = \frac{\sqrt[4]{81 \cdot 2}}{\sqrt[4]{16 \cdot 5}} = \frac{\sqrt[4]{81} \ \cdot \sqrt[4]{2}} {\sqrt[4]{16} \ \cdot \sqrt[4]{5}} = \frac{3 \sqrt[4]{2}}{2 \sqrt[4]{5}}\\ \text{Recombine the fraction under one radical sign:} && & = \frac{3}{2} \sqrt[4]{\frac{2}{5}}\end{align*}$$
  ::重写两个基的商数: 416280=4162480 分别简化每个基数: =48124165=481 42416445=342245 在一个基号下分数: =32425

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  c)  $\begin{align*}\sqrt[3]{135}\end{align*}$
  :c) 3135

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  Here we are looking for the product of the highest perfect cube root and another number. Often it’s not very easy to identify the perfect root in the expression under the radical sign. In this case, we can factor the number under the radical sign completely by using a factor tree:
  ::我们在这里寻找的是最高完美立方根和另一个数字的产物。 在激进标志下表达的完美根往往不容易辨别出来。 在这种情况下,我们可以通过使用要素树来将激进标志下的数字完全计入:

  

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  We see that $\begin{align*}135 = 3 \cdot 3 \cdot 3 \cdot 5 = 3^3 \cdot 5\end{align*}$ . Therefore $\begin{align*}\sqrt[3]{135} = \sqrt[3]{3^3 \cdot 5} = \sqrt[3]{3^3} \ \cdot \sqrt[3]{5} = 3 \sqrt[3]{5}\end{align*}$ .
  ::我们看到135=33335=335。 因此,3135=3335=333=3□35。

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  Now let’s see some examples involving variables.
  ::现在让我们来看看一些涉及变数的例子。

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  Examples
  ::实例

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

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  Treat constants and each variable separately and write each expression as the products of a perfect power as indicated by the index of the radical and another number.
  ::将常数和每个变量分开处理,并将每个表达式写成一个完美功率的产物,如激进和另一个数字的指数所示。

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  Example 1
  ::例1

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  $\begin{align*}\sqrt{12x^3 y^5}\end{align*}$

  $$\begin{align*}\text{Re-write as a product of radicals:} \sqrt{12x^3y^5} = \sqrt{12} \ \cdot \sqrt{x^3} \ \cdot \sqrt{y^5}\\ \text{Simplify each radical separately:} \left(\sqrt{4 \cdot 3}\right ) \cdot \left( \sqrt{x^2 \cdot x}\right ) \cdot \left (\sqrt{y^4 \cdot y}\right ) = \left (2 \sqrt{3}\right ) \cdot \left (x \sqrt{x}\right ) \cdot \left (y^2 \sqrt{y}\right )\\ \text{Combine all terms outside and inside the radical sign:} =2xy^2 \sqrt{3xy}\end{align*}$$
  ::12x3y5 re- write as a bigs 的产物 : 12x3y512 3 y5 5 分别简化每个基数 : (43) (x2xx) (y4y) = (23) (xxx) (y2y) (y2y) (y2y) 将基号内外的所有条件结合起来 : =2xy23xy

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  Example 2
  ::例2

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  $\begin{align*}\sqrt[4]{\frac{1250x^7}{405y^9}}\end{align*}$

  $$\begin{align*}\text{Re-write as a quotient of radicals:} \sqrt[4]{\frac{1250x^7}{405y^9}} = \frac{\sqrt[4]{1250x^7}}{\sqrt[4]{405y^9}}\\ \text{Simplify each radical separately:} = \frac{\sqrt[4]{625 \cdot 2} \ \cdot \sqrt[4]{x^4 \cdot x^3}}{\sqrt[4]{81 \cdot 5} \ \cdot \sqrt[4]{y^4 \cdot y^4 \cdot y}} = \frac{5 \sqrt[4]{2} \cdot x \cdot \sqrt[4]{x^3}}{3 \sqrt[4]{5} \cdot y \cdot y \ \cdot \sqrt[4]{y}} = \frac{5x \sqrt[4]{2x^3}}{3y^2 \sqrt[4]{5y}}\\ \text{Recombine fraction under one radical sign:} = \frac{5x}{3y^2} \sqrt[4]{\frac{2x^3}{5y}}\end{align*}$$
  ::41250x7405y9 重写为基数: 41250x7405y9=41250x74}4405y9 将每个基数分别简化: =46252 44x4x4x34}3}815 444yyyyy=5422×x4}4}4}3}3}3}544y=5x42x33y24}5yRecombine 碎片,一个基号:=5x3y24}2x35y

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  Review 
  ::回顾

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

  1. $\begin{align*}\sqrt{169}\end{align*}$
  2. $\begin{align*}\sqrt[4]{-81}\end{align*}$
  3. $\begin{align*}\sqrt[3]{-125}\end{align*}$
  4. $\begin{align*}\sqrt[5]{1024}\end{align*}$

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  Write each expression as a rational exponent.
  ::将每个表达式写成理性的引言 。

  5. $\begin{align*}\sqrt[3]{14}\end{align*}$
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  6. $\begin{align*}\sqrt[4]{zw}\end{align*}$
     ::4zw
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  7. $\begin{align*}\sqrt{a}\end{align*}$
     ::a
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  8. $\begin{align*}\sqrt[9]{y^3}\end{align*}$
     ::9y3

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

  9. $\begin{align*}\sqrt{24}\end{align*}$
  10. $\begin{align*}\sqrt{300}\end{align*}$
  11. $\begin{align*}\sqrt[5]{96}\end{align*}$
  12. $\begin{align*}\sqrt{\frac{240}{567}}\end{align*}$
  13. $\begin{align*}\sqrt[3]{500}\end{align*}$
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  14. $\begin{align*}\sqrt[6]{64x^8}\end{align*}$
      ::664x8
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  15. $\begin{align*}\sqrt[3]{48a^3 b^7}\end{align*}$
      ::348a3b7
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  16. $\begin{align*}\sqrt[3]{\frac{16x^5}{135y^4}}\end{align*}$
      ::316x5135y4

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  Review (Answers)
  ::回顾(答复)

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  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 。