6.3 指数的权力属性

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  指数的功率属性

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  There are 1,000 bacteria present in a culture. When the culture is treated with an antibiotic, the bacteria count is halved every 4 hours. How many bacteria remained 24 hours later?
  ::文化中存在1000种细菌。当文化被用抗生素处理时,细菌数每4小时减半。24小时后还剩下多少细菌?

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  Power Properties of Exponents
  ::指数的功率属性

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  The last set of properties to explore are the power properties. Let’s investigate what happens when a power is raised to another power.
  ::最后一组要探索的属性是功率属性。让我们来调查当一个权力被提升到另一个权力时会发生什么。

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  Power of a Power Property
  ::权力财产权

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  Step 1: Rewrite $\begin{align*}(2^3)^5\end{align*}$ as $\begin{align*}2^3\end{align*}$ five times.
  ::第1步:重写(23)5为23次5次。

  $\begin{align*}(2^3)^5 = 2^3 \cdot 2^3 \cdot 2^3 \cdot 2^3 \cdot 2^3\end{align*}$

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  Step 2: Expand each $\begin{align*}2^3\end{align*}$ . How many 2’s are there?
  ::第二步:扩大每个23个,有多少2个?

  $\begin{align*}(2^3)^5 = \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} \cdot \underbrace{2 \cdot 2 \cdot 2}_{2^3} = 2^{15}\end{align*}$

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  Step 3: What is the product of the powers?
  ::第3步:权力的产物是什么?

  $\begin{align*}3 \cdot 5 = 15\end{align*}$

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  Step 4: Fill in the blank. $\begin{align*}(a^m)^n = a^{-^\cdot-}\end{align*}$
  ::第4步:填空。 (am)n=a

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  $\begin{align*}(a^m)^n = a^{mn}\end{align*}$
  :am)n=amn(上午)

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  The other two exponent properties are a form of the .
  ::其他两种指数属性是...的一种形式。

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  Power of a Product Property : $\begin{align*}(ab)^m = a^m b^m\end{align*}$
  ::产品产权的功率ab)m=ambm

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  Power of a Quotient Property : $\begin{align*}\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}\end{align*}$
  ::引号属性的功率ab)m=ambm

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  Simplify $\begin{align*}(3^4)^2\end{align*}$
  ::简化( 3434) 2

  $$\begin{align*}(3^4)^2 = 3^{4 \cdot 2} = 3^8 = 6561\end{align*}$$

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  Simplify $\begin{align*}(x^2 y)^5\end{align*}$
  ::简化 (x2y) 5

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  $$\begin{align*}(x^2 y)^5 = x^{2 \cdot 5} y^5 = x^{10} y^5\end{align*}$$
  :x2y) 5=x2=x2=5y5=x10y5

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  Simplify $\begin{align*}\left( \frac{3a^{-6}}{2^2 a^2} \right)^4\end{align*}$ (do not leave any negative exponents)
  ::简化 (3a- 622a2) 4 (不要留下任何负指数)

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  This problem  uses the Negative Exponent Property . Distribute the $\begin{align*}4^{th}\end{align*}$ power first and then move the negative power of $\begin{align*}a\end{align*}$ from the numerator to the denominator.
  ::这个问题使用负指数属性。 先分配第4次功率, 然后将分子的负功率从分子移动到分母 。

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  $$\begin{align*}\left( \frac{3a^{-6}}{2^2 a^2} \right)^4 = \frac{3^4 a^{-6 \cdot 4}}{2^{2 \cdot 4} a^{2 \cdot 4}} = \frac{81a^{-24}}{2^8 a^8} = \frac{81}{256a^{8+24}} = \frac{81}{256a^{32}}\end{align*}$$
  :3a-622a2)4=34a-64224a24=81a-2428a8=81256a8+24=81256a32

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  Simplify $\begin{align*}\frac{4x^{-3} y^4 z^6}{12x^2 y} \div \left( \frac{5xy^{-1}}{15x^3 z^{-2}} \right)^2\end{align*}$  (do not leave any negative exponents)
  ::简化 4x- 3y4z4z612x2y( 5xy- 115x3z-2) 2 (不要留下任何负指数)

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  This problem is definitely as complicated as these types of problems get. Here, all the properties of exponents will be used. Remember that dividing by a fraction is the same as multiplying by its reciprocal .
  ::这个问题肯定和这类问题一样复杂。 在这里, 所有的指数属性都将被使用。 记住, 分数除以一个分数与对等乘法相同 。

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  $$\begin{align*}\frac{4x^{-3} y^4 z^6}{12x^2 y} \div \left( \frac{5xy^{-1}}{15x^3 z^{-2}} \right)^2 &= \frac{4x^{-3} y^4 z^6}{12x^2 y} \cdot \frac{225x^6 z^{-4}}{25x^2 y^{-2}}\\ &= \frac{y^3 z^6}{3x^5} \cdot \frac{9x^4 y^2}{z^4}\\ &= \frac{3x^4 y^5 z^6}{x^5 z^4}\\ &= \frac{3y^5 z^2}{x}\end{align*}$$
  ::4-3y4z612x2y(5xy-115x3z-2)2=4x-3y4z612x2y225x6z-425x2y=425x2y2=y3z63x5}9x4y2z4=3x4y5z6x5z4=3y5z2x

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  Examples
  ::实例

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  Example 1
  ::例1

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  Earlier, you were asked to find the number of  bacteria that remained 24 hours later. 
  ::早些时候,有人要求你找到24小时后留下的细菌数量。

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  To find the number of bacteria remaining, we use the exponential expression $\begin{align*}1000 (\frac{1}{2})^n\end{align*}$ where n is the number of four-hour periods.
  ::为了找到剩余细菌的数量, 我们使用指数表达式 1000( 12)n, 其中 n 是四个小时的间隔数 。

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  There are 6 four-hour periods in 24 hours, so we set n equal to 6 and solve.
  ::24小时有6个4小时的时段 所以我们设定了等于6的时段 然后解决

  $\begin{align*}1000 (\frac{1}{2})^6\end{align*}$

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  Applying the Power of a Quotient Property, we get:
  ::运用引号属性的力量,我们得到:

  $\begin{align*}1000 (\frac{1^6}{2^6}) = \frac {1000 \cdot 1}{2^6} = \frac {1000}{64} = 15.625\end{align*}$

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  Therefore , there are 15.625 bacteria remaining after 24 hours.
  ::因此,24小时后还剩下15.625种细菌。

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  Example 2
  ::例2

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  Simplify without negative exponents:  $\begin{align*}\left( \frac{5a^3}{b^4} \right)^7\end{align*}$ .
  :5a3b4)7。

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  Distribute the 7 to every power within the parenthesis.
  ::将7号分配到括号内的所有权力

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  $$\begin{align*}\left( \frac{5a^3}{b^4} \right)^7 = \frac{5^7 a^{21}}{b^{28}} = \frac{78,125a^{21}}{b^{28}}\end{align*}$$
  :5a3b4)7=57a21b28=78,125a21b28

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  Example 3
  ::例3

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  Simplify without negative exponents:  $\begin{align*}(2x^5)^{-3} (3x^9)^2\end{align*}$ .
  ::简化无负指数2x5)-3(3x9)2。

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  Distribute the -3 and 2 to their respective parenthesis and then use the properties of negative exponents, quotient and product properties to simplify.
  ::将 - 3 和 2 分配给各自的括号,然后使用负指数、商数和产品属性的特性进行简化。

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  $$\begin{align*}(2x^5)^{-3} (3x^9)^2 = 2^{-3} x^{-15} 3^2 x^{18} = \frac{9x^3}{8}\end{align*}$$
  :2x5)-3(3x9)2=2-3x-1532x18=9x38)

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  Example 4
  ::例4

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  Simplify without negative exponents:  $\begin{align*}\frac{(5x^2 y^{-1})^3}{10y^6} \cdot \left( \frac{16x^8 y^5}{4x^7} \right)^{-1}\end{align*}$ .
  ::简化时没有负引号 : (5x2y- 1) 310y6}(16x8y54x7) - 1 。

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  Distribute the exponents that are outside the parenthesis and use the other properties of exponents to simplify. Anytime a fraction is raised to the -1 power, it is equal to the reciprocal of that fraction to the first power.
  ::分配括号外的推数, 并使用推数的其他属性来简化。 每当一个分数被提升到 - 1 功率时, 它等于该分数与第一个功率的对等值 。

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  $$\begin{align*}\frac{\left(5x^2 y^{-1}\right)^3}{10y^6} \cdot \left( \frac{16x^8 y^5}{4x^7} \right)^{-1} &= \frac{5^3 x^{-6} y^{-3}}{10y^6} \cdot \frac{4x^7}{16x^8 y^5}\\ &= \frac{500xy^{-3}}{160x^8 y^{11}}\\ &= \frac{25}{8x^7 y^{14}}\end{align*}$$
  :5x2y-1)310y6(16x8y54x7)-1=53x-6y-310y64x716x8y5=500xy-3160x8y11=258x7y14)

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  Review
  ::回顾

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  Simplify the following expressions without negative exponents.
  ::简化以下表达式,无负引号。

  1. $\begin{align*}(2^5)^3\end{align*}$
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  2. $\begin{align*}(3x)^4\end{align*}$
     :3x)4
  3. $\begin{align*}\left( \frac{4}{5} \right)^2\end{align*}$
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  4. $\begin{align*}(6x^3)^3\end{align*}$
     :6x3)3
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  5. $\begin{align*}\left( \frac{2a^3}{b^5} \right)^7\end{align*}$
     :2a3b5)7
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  6. $\begin{align*}(4x^8)^{-2}\end{align*}$
     :4x8)-2
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  7. $\begin{align*}\left( \frac{1}{7^2 h^9} \right)^{-1}\end{align*}$
     :172h9)-1
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  8. $\begin{align*}\left( \frac{2x^4 y^2}{5x^{-3} y^5} \right)^3\end{align*}$
     :2x4y25x-3y5)3
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  9. $\begin{align*}\left( \frac{9m^5 n^{-7}}{27 m^6 n^5} \right)^{-4}\end{align*}$
     :9m5n-7727m6n5)-4
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  10. $\begin{align*}\frac{(4x)^2 (5y)^{-3}}{(2x^3 y^5)^2}\end{align*}$
      :4x)2(5y)-3(2x3y5)2
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  11. $\begin{align*}(5r^6)^4 \left( \frac{1}{3} r^{-2} \right)^5\end{align*}$
      :5r6)4(13r-2)5
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  12. $\begin{align*}(4t^{-1} s)^3 (2^{-1} ts^{-2})^{-3}\end{align*}$
      :4吨-1s)3(2吨-1吨-2)-3
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  13. $\begin{align*}\frac{6a^2 b^4}{18a^{-3} b^4} \cdot \left( \frac{8b^{12}}{40a^{-8} b^5} \right)^2\end{align*}$
      ::6a2b418a-3b4(8b1240a-8b5)2
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  14. $\begin{align*}\frac{2(x^4 y^4)^0}{2^4 x^3 y^5 z} \div \frac{8z^{10}}{32x^{-2} y^5}\end{align*}$
      ::2(x4y4)024x3y5z8z1032x-2y5
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  15. $\begin{align*}\frac{5g^6}{15g^0 h^{-1}} \cdot \left( \frac{h}{9g^{15} j^7} \right)^{-3}\end{align*}$
      ::5g615g0h-1(h9g15j7)-3
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  16. Challenge  $\begin{align*}\frac{a^7 b^{10}}{4a^{-5} b^{-2}} \cdot \left[ \frac{(6ab^{12})^2}{12a^9 b^{-3}} \right]^2 \div (3a^5 b^{-4})^3\end{align*}$
      ::a7b104a-5b-2[(6ab12)21212a9b-33]2(3a5b-4)3
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  17. Rewrite $\begin{align*}4^3\end{align*}$ as a power of 2.
      ::重写43是2的功率
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  18. Rewrite $\begin{align*}9^2\end{align*}$ as a power of 3.
      ::将92重写为3的功率
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  19. Solve the equation for $\begin{align*}x\end{align*}$ . $\begin{align*}3^2 \cdot 3^x = 3^8\end{align*}$
      ::解析 x. 323x=38 的方程
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  20. Solve the equation for $\begin{align*}x\end{align*}$ . $\begin{align*}(2^x)^4 = 4^8\end{align*}$
      ::解析 x. (2x)4=48 的方程

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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 。