节: 定义 nth 根 | 7.1 界定nth根

章节大纲

The volume of a cube is found to be $\begin{aligned} 343 s^{7} \end{aligned}$ . What is the length of each side of the cube?
::立方体的体积为343.7。立方体每一侧的长度是多少?

nth Roots
::nth 根

So far, we have seen exponents with integers and the square root . In this concept, we will link roots and exponents. First, let’s define additional roots. Just like the square and the square root are inverses of each other, the inverse of a cube is the cubed root . The inverse of the fourth power is the fourth root.
::到目前为止,我们已经看到以整数和平方根的表情。在这个概念中,我们将连接根和直方根。首先,让我们来定义更多的根。就像方根和平方根是反向的一样,立方体的反面是立方根。第四个功率的反面是第四根。

$\begin{aligned} \sqrt[3]{27} = \sqrt[3]{3^{3}} = 3, \sqrt[5]{32} = \sqrt[5]{2^{5}} = 2 \end{aligned}$

The $\begin{aligned} n^{t h} \end{aligned}$ root of a number, $\begin{aligned} x^{n} \end{aligned}$ , is $\begin{aligned} x, \sqrt[n]{x^{n}} = x \end{aligned}$ . And, just like simplifying square roots, we can simplify $\begin{aligned} n^{t h} \end{aligned}$ roots.
::数字的 nthroot, xn, 是 x, xnn=x。就像简化平方根一样, 我们可以简化 nth root 。

Let's find $\begin{aligned} \sqrt[6]{729} \end{aligned}$ .
::我们去找7296

To simplify a number to the sixth root, there must be 6 of the same factor to pull out of the root.
::要将数字简化为第六个根,必须有一个因素中的六个因素才能从根中拔出。

$\begin{aligned} 729 = 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 3^{6} \end{aligned}$

Therefore , $\begin{aligned} \sqrt[6]{729} = \sqrt[6]{3^{6}} = 3 \end{aligned}$ . The sixth root and the sixth power cancel each other out. We say that 3 is the sixth root of 729.
::因此, 7296=366=3. 第六根和第六电源互相取消。我们说, 3是第六根 729。

From this problem , we can see that it does not matter where the exponent is placed, it will always cancel out with the root.
::从这个问题中,我们可以看到,不管出价放在哪里,它总是用根来取消。

$\begin{aligned} \sqrt[6]{3^{6}} & = {\sqrt[6]{3}}^{6} o r {(\sqrt[6]{3})}^{6} \\ \sqrt[6]{729} & = {(1.2009 \dots)}^{6} \\ 3 & = 3 \end{aligned}$

::366=366或3667296=(1.2009...)63=3

So, it does not matter if you evaluate the root first or the exponent.
::所以,无论你先评价根还是先评价根,都无关紧要。

The $\begin{aligned} n^{t h} \end{aligned}$ Root Theorem : For any real number $\begin{aligned} a \end{aligned}$ , root $\begin{aligned} n \end{aligned}$ , and exponent $\begin{aligned} m \end{aligned}$ , the following is always true: $\begin{aligned} \sqrt[n]{a^{m}} = {\sqrt[n]{a}}^{m} = {(\sqrt[n]{a})}^{m} \end{aligned}$ .
::Nth 根理论: 对于任何真正的数字 a, root n, 和表率 m, 以下总是真实的: amn=anm=( an) m 。

Evaluate: $\begin{aligned} \sqrt[5]{32^{3}} \end{aligned}$
::评价:3235

If you solve this problem as written, you would first find $\begin{aligned} 32^{3} \end{aligned}$ and then apply the $\begin{aligned} 5^{t h} \end{aligned}$ root.
::如果您以书面方式解决这个问题, 将首先找到 323, 然后应用 5 根。

$\begin{aligned} \sqrt[5]{32^{3}} = \sqrt[5]{38768} = 8 \end{aligned}$

However, this would be very difficult to do without a calculator. This is an example where it would be easier to apply the root and then the exponent. Let’s rewrite the expression and solve.
::然而,如果没有计算器,这很难做到。这是一个比较容易应用根和推手的例子。让我们重写表达式和解答。

$\begin{aligned} {\sqrt[5]{32}}^{3} = 2^{3} = 8 \end{aligned}$

Evaluate: $\begin{aligned} {\sqrt{16}}^{3} \end{aligned}$
::评价:163

This problem does not need to be rewritten. $\begin{aligned} \sqrt{16} = 4 \end{aligned}$ and then $\begin{aligned} 4^{3} = 64 \end{aligned}$ .
::这个问题不需要重写。 16=4, 然后43=64。

Finally, let's simplify the following.
::最后,让我们简化以下内容。

Evaluate: $\begin{aligned} \sqrt[4]{64} \end{aligned}$
::评价:644

To simplify the fourth root of a number, there must be 4 of the same factor to pull it out of the root. Let’s write the prime factorization of 64 and simplify.
::要简化数字的第四根根,必须用同一种因素中的4根来将其从根中拉出来。让我们写64的质因数并简化。

$\begin{aligned} \sqrt[4]{64} = \sqrt[4]{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2} = 2 \sqrt[4]{4} \end{aligned}$

Notice that there are 6 2’s in 64. We can pull out 4 of them and 2 2’s are left under the radical .
::我们可以撤走其中4个, 2个被留在激进组织之下。

Evaluate: $\begin{aligned} \sqrt[3]{\frac{54 x^{3}}{125 y^{5}}} \end{aligned}$
::评价:54x3125y53

Just like simplifying fractions with square roots, we can separate the numerator and denominator.
::就像简化有平方根的分数一样, 我们可以分离分子和分母。

$\begin{aligned} \sqrt[3]{\frac{54 x^{3}}{125 y^{5}}} = \frac{\sqrt[3]{54 x^{3}}}{\sqrt[3]{125 y^{5}}} = \frac{\sqrt[3]{2 \cdot 3 \cdot 3 \cdot 3 \cdot x^{3}}}{\sqrt[3]{5 \cdot 5 \cdot 5 \cdot y^{3} \cdot y^{2}}} = \frac{3 x \sqrt[3]{2}}{5 y \sqrt[3]{y^{2}}} \end{aligned}$
::54x3125y53 = 54x33125y53 = 2}3}3}3}3}3×3}3}3}3}3}3}3}3}3}3}5}5}5}5}}}y}3}23=3×23}3×23}23

Notice that because the $\begin{aligned} x \end{aligned}$ is cubed, the cube and cubed root cancel each other out. With the $\begin{aligned} y \end{aligned}$ - term , there were five, so three cancel out with the root, but two are still left under radical.
::请注意, 因为 x 是立方体, 立方体和立方根会相互取消。在 Y 期, 有五个, 所以有三个取消根, 但两个仍然在基下。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the length of each side of the cube.
::早些时候,你被要求找到立方体每一侧的长度。

Recall that the volume of a cube is $\begin{aligned} V = s^{3} \end{aligned}$ , where s is the length of each side. So to find the side length, take the cube root of $\begin{aligned} 343 z^{7} \end{aligned}$ .
::回顾立方体的体积是 V=3, 方体的长度为每边的长度。要找到侧长, 请选择343z7 的立方体根。

First, you can separate this number into two different roots, $\begin{aligned} \sqrt[3]{343} \cdot \sqrt[3]{z^{7}} \end{aligned}$ . Now, simplify each root.
::首先,您可以将这个数字分为两个不同的根,3433z73。现在,简化每个根。

$\begin{aligned} \sqrt[3]{343} \cdot \sqrt[3]{z^{7}} = \sqrt[3]{7^{3}} \cdot \sqrt[3]{z^{3} \cdot z^{3} \cdot z} = 7 z^{2} \sqrt[3]{z} \end{aligned}$
::3433z73=733z3z3z3z3=7z2z3

Therefore, the length of the cube's side is $\begin{aligned} 7 z^{2} \sqrt[3]{z} \end{aligned}$ .
::因此,立方体侧的长度是7z2z3。

Example 2
::例2

Simplify: $\begin{aligned} \sqrt[4]{625 z^{8}} \end{aligned}$
::简化: 625z84

First, you can separate this number into two different roots, $\begin{aligned} \sqrt[4]{625} \cdot \sqrt[4]{z^{8}} \end{aligned}$ . Now, simplify each root.
::首先,您可以将这个数字分为两个不同的根, 6254z84。现在, 简化每个根。

$\begin{aligned} \sqrt[4]{625} \cdot \sqrt[4]{z^{8}} = \sqrt[4]{5^{4}} \cdot \sqrt[4]{z^{4} \cdot z^{4}} = 5 z^{2} \end{aligned}$
::6254z84=544z4z44=5z2

When looking at the $\begin{aligned} z^{8} \end{aligned}$ , think about how many $\begin{aligned} z^{4} \end{aligned}$ you can even pull out of the fourth root. The answer is 2, or a $\begin{aligned} z^{2} \end{aligned}$ , outside of the radical.
::当看Z8时, 想想有多少Z4可以从第四根中提取出来。答案是 2 或 z2 , 在激进分子之外。

Example 3
::例3

Simplify: $\begin{aligned} \sqrt[7]{32 x^{5} y} \end{aligned}$
::简化: 32x5y7

$\begin{aligned} 32 = 2^{5} \end{aligned}$ , which means there are not 7 2's that can be pulled out of the radical. Same with the $\begin{aligned} x^{5} \end{aligned}$ and the $\begin{aligned} y \end{aligned}$ . Therefore, you cannot simplify the expression any further.
::32=25, 这意味着没有 7 2 值可以从基体中提取出来。与 x5 和 y 相同。因此, 您无法进一步简化表达式。

Example 4
::例4

Simplify: $\begin{aligned} \sqrt[5]{9216} \end{aligned}$
::简化:92165

Write out 9216 in the prime factorization and place factors into groups of 5.
::在初级系数中写9216,并将各种因素分为5组。

$\begin{aligned} \sqrt[5]{9216} & = \sqrt[5]{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3} \\ = \sqrt[5]{2^{5} \cdot 2^{5} \cdot 3^{2}} \\ = 2 \cdot 2 \sqrt[5]{3^{2}} \\ = 4 \sqrt[5]{9} \end{aligned}$

Example 5
::例5

Simplify: $\begin{aligned} \sqrt[3]{\frac{40}{175}} \end{aligned}$
::简化: 401753

Reduce the fraction , separate the numerator and denominator and simplify.
::减少分数,分离分子和分母并简化。

$\begin{aligned} \sqrt[3]{\frac{40}{175}} = \sqrt[3]{\frac{8}{35}} = \frac{\sqrt[3]{2^{3}}}{\sqrt[3]{35}} = \frac{2}{\sqrt[3]{35}} \cdot \frac{\sqrt[3]{35^{2}}}{\sqrt[3]{35^{2}}} = \frac{2 \sqrt[3]{1225}}{35} \end{aligned}$

In the red step, we rationalized the denominator by multiplying the top and bottom by $\begin{aligned} \sqrt[3]{35^{2}} \end{aligned}$ , so that the denominator would be $\begin{aligned} \sqrt[3]{35^{3}} \end{aligned}$ or just 35. Be careful when rationalizing the denominator with higher roots!
::在红色步骤中,我们通过将顶部和底部乘以3523, 使分母合理化, 这样分母就会是3533或35。当将分母合理化时要小心高根!

Review
::回顾

Reduce the following radical expressions.
::减少以下激进表达式。

$\begin{aligned} \sqrt[3]{81} \end{aligned}$

$\begin{aligned} {\sqrt[4]{625}}^{3} \end{aligned}$

$\begin{aligned} \sqrt{9^{5}} \end{aligned}$

$\begin{aligned} \sqrt[5]{128} \end{aligned}$

$\begin{aligned} \sqrt{\sqrt{10000}} \end{aligned}$

$\begin{aligned} {\sqrt{\frac{25}{8}}}^{4} \end{aligned}$

$\begin{aligned} {\sqrt[6]{64}}^{5} \end{aligned}$

$\begin{aligned} {\sqrt[3]{\frac{8}{81}}}^{2} \end{aligned}$

$\begin{aligned} \sqrt[4]{\frac{243}{16}} \end{aligned}$

$\begin{aligned} \sqrt[3]{24 x^{5}} \end{aligned}$
::24x53 24x53

$\begin{aligned} \sqrt[4]{48 x^{7} y^{13}} \end{aligned}$
::48x7y134

$\begin{aligned} \sqrt[5]{\frac{160 x^{8}}{y^{7}}} \end{aligned}$
::160x88y75

$\begin{aligned} {\sqrt[3]{1000 x^{6}}}^{2} \end{aligned}$
::1000x632

$\begin{aligned} \sqrt[4]{\frac{162 x^{5}}{y^{3} z^{10}}} \end{aligned}$
::162x5y3z104

$\begin{aligned} {\sqrt{40 x^{3} y^{4}}}^{3} \end{aligned}$
::40x3y43

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

nth Roots ::nth 根

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

nth Roots
::nth 根

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)