Course: 7.2 理性指数和根

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理性指数和根
A planet's maximum distance from the sun (in astronomical units) is given by the formula $\begin{align*}d = p^{\frac {2}{3}}\end{align*}$ , were 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 给出了行星与太阳的最大距离。如果行星绕太阳的轨道是27年,那么它与太阳的距离是多少?

The Rational Exponent Theorem
::理性指数定理

Now that you are familiar with , we will convert them into exponents. Let’s look at the square root and see if we can use the properties of exponents to determine what exponential number it is equivalent to.
::既然您已经熟悉了 , 我们将把它们转换成指数。让我们看看平方根, 看看我们能否使用指数的属性来确定指数数的等值。

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

Step 1: Evaluate $\begin{align*}\left(\sqrt{x}\right)^2\end{align*}$ . What happens?
::步骤1:评价(x)2. 会发生什么?

The $\begin{align*}\sqrt{\;\;}\end{align*}$ and the $\begin{align*}^2\end{align*}$ cancel each other out, $\begin{align*}\left(\sqrt{x}^2\right)=x\end{align*}$ .
::和2 互相取消对方出,(x2)=x。

Step 2: Recall that when a power is raised to another power, we multiply the exponents. Therefore , we can rewrite the exponents and root as an equation , $\begin{align*}n\cdot 2=1\end{align*}$ . Solve for $\begin{align*}n\end{align*}$ .
::步骤2:回顾当电力被提升到另一电力时,我们乘以指数。因此,我们可以将指数和根重写为等式,n2=1. 解决n。

$\begin{aligned} \frac{n \cdot 2}{2} & = \frac{1}{2} \\ n & = \frac{1}{2} \end{aligned}$
$\begin{align*}\frac{n \cdot \cancel{2}}{\cancel{2}}&=\frac{1}{2} \\ n&=\frac{1}{2}\end{align*}$
::n22=12n=12

Step 3: From Step 2, we can conclude that $\begin{align*}\sqrt{\;\;}=\frac{1}{2}\end{align*}$ .
::步骤3:从步骤2开始,我们可以得出=12的结论。

$\begin{align*}\left(\sqrt{x}\right)^2=\left(x^{\frac{1}{2}}\right)^2=x^{\left(\frac{1}{2}\right)\cdot 2}=x^1=x\end{align*}$
:x)2=(x12)2=x(12)2=x(x)2=x1=x

From the steps above, we see that $\begin{align*}\sqrt{x}=x^{\frac{1}{2}}\end{align*}$ . We can extend this idea to the other roots as well; $\begin{align*}\sqrt[3]{x}=x^{\frac{1}{3}}=\sqrt[4]{x}=x^{\frac{1}{4}},\ldots \sqrt[n]{x}=x^{\frac{1}{n}}\end{align*}$ .
::从上面的阶梯可以看到 x=x12。我们可以将这个想法扩展至其它根; x3=x13=x4=x14,...xn=x1n。

The Rational Exponent Theorem : For any real number $\begin{align*}a\end{align*}$ , root $\begin{align*}n\end{align*}$ , and exponent $\begin{align*}m\end{align*}$ , the following is always true: $\begin{align*}a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m\end{align*}$ .
::理性指数理论:对于任何真正的数字、根和指数,以下总是真实的:amn=amn=(an)m。

Find $\begin{align*}256^{\frac{1}{4}}\end{align*}$ .
::找25614号

Rewrite this expression in terms of roots. A number to the one-fourth power is the same as the fourth root.
::以根重重写此表达式。一至四位数与四根数相同。

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

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

Find $\begin{align*}49^{\frac{3}{2}}\end{align*}$ .
::找4932号

This problem is similar to ones you have seen already. However, now, the root is written in the exponent. Rewrite the problem.
::这个问题与您已经看到的问题相似。但是, 现在, 根写在提示中。重写问题。

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

I t is easier to evaluate the second option above.
::比较容易评价上述第二种选择。

So $\begin{align*}\left(\sqrt{49}\right)^3=7^3=343\end{align*}$ .
::所以(493=73=343)

Find $\begin{align*}5^{\frac{2}{3}}\end{align*}$ using a calculator. Round your answer to the nearest hundredth.
::使用计算器查找 523 。将您的答复回圆到最近的第一百次。

To type this into a calculator, the keystrokes would probably look like: $\begin{align*}5^{\frac{2}{3}}\end{align*}$ . The “^” symbol is used to indicate a power. Anything in parenthesis after the “^” would be in the exponent. Evaluating this, we have 2.924017738..., or just 2.92.
::将此输入计算器时, 键盘可能会看起来像 : 523 。 “ ” 符号用于表示一种力量。 “ ” 之后的任何括号都会在提示中。评估一下, 我们有2. 92401778... 或2. 92。

Other calculators might have a $\begin{align*}x^y\end{align*}$ button. This button has the same purpose as the ^ and would be used in the exact same way.
::其它计算器可能有一个 Xy 按钮。此按钮的目的与 {} 相同, 并且将以同样的方式使用。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the planet's distance from the sun.
::早些时候,有人要求你找到地球与太阳的距离

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

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

Rewrite the problem.
::重写问题。

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

$\begin{align*}\left(\sqrt[3]{27}\right)^2=3^2=9\end{align*}$ .

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 2
::例2

Rewrite $\begin{align*}\sqrt[7]{12}\end{align*}$ using rational exponents. Then, use a calculator to find the answer.
::使用理性引言重写 127 。然后, 使用计算器来找到答案。

Using rational exponents, the $\begin{align*}7^{th}\end{align*}$ root becomes the $\begin{align*}\frac{1}{7}\end{align*}$ power; $\begin{align*}12^{\frac{1}{7}}=1.426\end{align*}$ .
::使用理性的指数, 第7根变成17功率; 1217=1.426。

Example 3
::例3

Rewrite $\begin{align*}845^{\frac{4}{9}}\end{align*}$ using roots. Then, use a calculator to find the answer.
::使用根重重写 84549。然后使用计算器找到答案。

Using roots, the 9 in the denominator of the exponent is the root; $\begin{align*}\sqrt[9]{845^4}=19.99\end{align*}$ . To enter this into a calculator, you can use the rational exponents. If you have a TI-83 or 84, press MATH and select 5 : $\begin{align*}\sqrt[x]{\;\;}\end{align*}$ . On the screen, you should type $\begin{align*}9\sqrt[x]{\;\;} \ 845^\land 4\end{align*}$ to get the correct answer. You can also enter $\begin{align*}845^\land {\left(\frac{4}{9}\right)}\end{align*}$ and get the exact same answer
::使用根, 指数分母中的 9 值是 root; 84549 = 19. 99。要将此输入计算器, 您可以使用理性推算器。如果您有 TI- 83 或 84, 请按 MATH 键并选择 5: x。在屏幕上, 您应该输入 9x 8454 4 以获得正确的答案。您也可以输入 845\\\\ (49) 并获得相同的答案

Example 4
::例4

Evaluate without a calculator: $\begin{align*}125^{\frac{4}{3}}\end{align*}$ .
::无计算器的评价:12543。

$\begin{align*} 125^{\frac{4}{3}}=\left(\sqrt[3]{125}\right)^4=5^4=625\end{align*}$

Example 5
::例5

Evaluate without a calculator: $\begin{align*}256^{\frac{5}{8}}\end{align*}$ .
::无计算法的评价:25658。

$\begin{align*}256^{\frac{5}{8}}=\left(\sqrt[8]{256}\right)^5=2^5=32\end{align*}$

Example 6
::例6

Evaluate without a calculator: $\begin{align*}\sqrt{81^{\frac{1}{2}}}\end{align*}$ .
::无计算法的评价:8112。

$\begin{align*}\sqrt{81^{\frac{1}{2}}}=\sqrt{\sqrt{81}}=\sqrt{9}=3\end{align*}$

Review
::回顾

Write the following expressions using rational exponents and then evaluate using a calculator. Answers should be rounded to the nearest hundredth.
::使用理性引言写下以下表达式,然后使用计算器进行评估。答案应四舍五入到最接近的百位。

$\begin{align*}\sqrt[5]{45}\end{align*}$

$\begin{align*}\sqrt[9]{140}\end{align*}$

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

Write the following expressions using roots and then evaluate using a calculator. Answers should be rounded to the nearest hundredth.
::使用根写下以下表达式,然后使用计算器进行评估。答案应四舍五入到最接近的百位。

$\begin{align*}72^{\frac{5}{3}}\end{align*}$

$\begin{align*}95^{\frac{2}{3}}\end{align*}$

$\begin{align*}125^{\frac{3}{4}}\end{align*}$

Evaluate the following without a calculator.
::在无计算器的情况下评价以下各点。

$\begin{align*}64^{\frac{2}{3}}\end{align*}$

$\begin{align*}27^{\frac{4}{3}}\end{align*}$

$\begin{align*}16^{\frac{5}{4}}\end{align*}$

$\begin{align*}\sqrt{25^3}\end{align*}$

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

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

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,请用合理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。

$\begin{align*}\sqrt[4]{\left(\frac{2}{3}\right)^8}\end{align*}$

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

$\begin{align*}\sqrt{\left(16\right)^{\frac{1}{2}}}^6\end{align*}$

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。

Section outline

General

理性指数和根

The Rational Exponent Theorem ::理性指数定理

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

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

81 1 2 = 81 = 9 = 3 \begin{align*}\sqrt{81^{\frac{1}{2}}}=\sqrt{\sqrt{81}}=\sqrt{9}=3\end{align*}

Review ::回顾

Review (Answers) ::回顾(答复)

The Rational Exponent Theorem
::理性指数定理

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

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

$\begin{align}\sqrt{81^{\frac{1}{2}}}=\sqrt{\sqrt{81}}=\sqrt{9}=3\end{align}$

Review
::回顾

Review (Answers)
::回顾(答复)