Section: 幂的评估与比较 | 1.9幂的评估与比较

Section outline

Su Chin keeps her turtle, Larry, in a glass tank that is 14 ³ cubic inches. Larry loves to eat and is growing at an alarming rate , and Su Chin will soon need a larger tank for him. She looks at an online pet store and sees that they have two tanks for sale: one is 2,744 cubic inches, and the other is 4,096 cubic inches. How can Su Chin compare the size of her tank to the new tanks to ensure she buys a larger one?
::苏钦把她的乌龟拉里留在一个143立方英寸的玻璃罐中。拉里喜欢吃,并且正在以惊人的速度增长,苏钦很快需要更大的坦克来为他服务。她看着一家在线宠物店,看到两辆坦克可以出售:一辆是2,744立方英寸,另一辆是4,096立方英寸。苏钦如何把她的坦克大小与新坦克相比,以确保她购买更大的坦克?

In this concept, you will learn how to evaluate and compare powers.
::在这个概念中,你将学会如何评估和比较权力。

Powers
::强权强权

$\begin{aligned} 4^{2} \end{aligned}$

Let's consider this expression.
::让我们考虑一下这个表达方式。

The large number, 4, is the base . The base is the number to be multiplied by itself.
::大数,4是基数。基数是自动乘以的数字。

The small number, 2, is the exponent . The exponent tells you how many times to multiply the base by itself. The exponent is also known as a power.
::数字小, 2 是一个指数。该指数告诉你多少次自己乘以基数。该指数也被称为一种力量。

Once you know the base and exponent, you can evaluate the expression. Evaluating an expression with an exponent means to complete the indicated multiplication and write the result as a product.
::一旦您知道该表达式的基数和表情,您就可以对表达式进行评价。用一个表情来评价表达式,以完成标注的乘数,并将结果写成产品。

Consider the expression above, $\begin{aligned} 4^{2} \end{aligned}$ .
::考虑上文的表述,4 2。

First, identify the base, in this case 4.
::首先,确定基地,在此情况下4。

Then, identify the exponent, in this case 2.
::然后,先确认前言者,在本案2中。

Next, write the expression as the base multiplied by itself the number of times indicated by the exponent:
::下一步,将表达式写成基数乘以指数显示的次数:

$\begin{aligned} 4^{2} = 4 \times 4 \end{aligned}$

Finally, perform the multiplication.
::最后,执行乘法。

$\begin{aligned} 4^{2} = 4 \times 4 = 16 \end{aligned}$

NOTE : A common error related to is to multiply the base by the exponent. Don't do that! Clearly, since $\begin{aligned} 4^{2} = 4 \times 4 = 16 \end{aligned}$ and $\begin{aligned} 4 \times 2 = 8 \end{aligned}$ , they aren't the same thing.
::注:一个常见的错误是将基数乘以指数。不要这样做! 显然, 由于 4 2 = 4 × 4 = 16 = 4 × 2 = 8, 它们不是一回事。

You can also compare the values of powers using symbols: $\begin{aligned} > \end{aligned}$ (greater than), $\begin{aligned} < \end{aligned}$ (less than) and $\begin{aligned} = \end{aligned}$ (equal to).
::您也可以使用符号来比较功率的值 : > (大于)、 < (小于) 和 = (等于) 。

To compare the values of different powers, you will need to evaluate each power and then compare them.
::为了比较不同力量的价值观,你需要评估每一力量,然后比较它们。

Let's look at an example.
::让我们举个例子。

Evaluate $\begin{aligned} 1^{100} \end{aligned}$
::评价 1 100

This looks a bit intimidating, since the power is so large. However, since the base is 1, and regardless how many times you multiply 1 by itself, it is always 1, this is actually not difficult at all.
::这看起来有点吓人,因为权力如此巨大。然而,由于基数是1,不管你自己乘乘1多少倍,这实际上并不困难。

The answer is: $\begin{aligned} 1^{100} = 1 \times 1 \times 1... \times 1 = 1 \end{aligned}$
::答案是: 1 100 = 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1 = 1

In fact, 1 to any power is equal to 1.
::事实上,一对一等于任何权力等于一。

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Su Chin and her rapidly growing reptile.
::早些时候,你被问及苏钦和她迅速成长的爬行动物的问题。

Su Chin needs to compare her current tank, which is 14 ³ cubic inches, with two possible replacements that measure 2,744 cubic inches, and 4,096 cubic inches, in order to choose the largest tank.
::Su Chin需要比较她目前的储油罐(143立方英寸)和两个可能的替代物(2,744立方英寸和4,096立方英寸),以便选择最大的储油罐。

First, evaluate the power of Su Chin’s current tank.
::首先,评估Su Chin目前坦克的威力。

14 ³ = 14 x 14 x 14 = 2,744
::143=14x14x14=2 744

The answer is 2,744 cubic inches.
::答案是2 744立方英寸

Then, compare those dimensions to the other tanks, using < , > , or =.
::然后,用 < 、 > 或 = 来将这些尺寸与其他罐体进行比较。

2,744 = 2,744

2,744 < 4,096

The answer is the second tank is largest.
::答案是第二辆坦克最大

The first tank is the same size as Su Chin’s current tank, at 2,744 cubic inches. The second tank is greater in size, at 4,096 cubic inches.
::第一个罐体的尺寸与Su Chin目前的罐体的尺寸相同,为2,744立方英寸。第二个罐体的尺寸更大,为4,096立方英寸。

Su Chin can now buy the second tank in order to upgrade Larry’s home to a spacious 4,096 cubic inches.
::Su Chin现在可以买第二辆坦克, 将Larry的家升级为4 096立方英寸宽的4 096立方英寸。

Example 2
::例2

Insert > , < or = in the blank.
::在空白中插入 > 、 < 或= 。

$\begin{aligned} 4^{5} \underline{} 5^{4} \end{aligned}$

First, evaluate each number.
::首先,评估每个数字。

$\begin{aligned} 4^{5} = & 4 \times 4 \times 4 \times 4 \times 4 \\ = & 16 \times 4 \times 4 \times 4 \\ = & 64 \times 4 \times 4 \\ = & 256 \times 4 \\ = & 1024 \\ 5^{4} = & 5 \times 5 \times 5 \times 5 \\ = & 25 \times 5 \times 5 \\ = & 125 \times 5 \\ = & 625 \end{aligned}$
Then, compare the values. 1024 > 625

The answer is $\begin{aligned} 4^{5} > 5^{4} \end{aligned}$
::答案是4 5 > 5 4

Example 3
::例3

Evaluate the expression $\begin{aligned} 2^{6} \end{aligned}$ .
::评估表达方式 2 6 。

First, identify the base, in this case 2
::首先,确定基准点,在此情况下2

Then, identify the exponent, in this case 6.
::然后,先确认出名者,在本案的6。

Next, consider the expression as the base multiplied by itself the number of times indicated by the exponent:
::接下来,将表达式视为基数乘以表情所示的次数:

$\begin{aligned} 2^{6} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \end{aligned}$
Finally, perform the multiplication.
$\begin{aligned} 2^{6} = & 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ = & 4 \times 2 \times 2 \times 2 \times 2 \\ = & 8 \times 2 \times 2 \times 2 \\ = & 16 \times 2 \times 2 \\ = & 32 \times 2 \\ = & 64 \end{aligned}$
The answer is 64.

Example 4
::例4

Evaluate the expression $\begin{aligned} 6^{3} \end{aligned}$ .
::评估表达方式 6 3

First, identify the base, in this case 6.
::首先,确定基数,在此情况下是6。

Then, identify the exponent, in this case 3.
::然后,确认前台,在此情况下,3。

Next, consider the expression as the base multiplied by itself the number of times indicated by the exponent:
::接下来,将表达式视为基数乘以表情所示的次数:

$\begin{aligned} 6^{3} = 6 \times 6 \times 6 \end{aligned}$
Finally, perform the multiplication.
$\begin{aligned} 6^{3} = & 6 \times 6 \times 6 \\ = & 36 \times 6 \\ = & 216 \end{aligned}$
The answer is 216.

Example 5
::例5

Which is greater? $\begin{aligned} 2^{7} \underline{} 5^{3} \end{aligned}$
::哪个更大? 2 7_ 5 3

First, evaluate each number.
::首先,评估每个数字。

$\begin{aligned} 2^{7} = & 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ = & 4 \times 2 \times 2 \times 2 \times 2 \times 2 \\ = & 8 \times 2 \times 2 \times 2 \times 2 \\ = & 16 \times 2 \times 2 \times 2 \\ = & 32 \times 2 \times 2 \\ = & 64 \times 2 \\ = & 128 \\ 5^{3} = & 5 \times 5 \times 5 \\ = & 25 \times 5 \\ = & 125 \end{aligned}$

Finally, compare the values. 128 > 125.
::最后,比较数值。 128 > 125。

The answer is $\begin{aligned} 2^{7} > 5^{3} \end{aligned}$ .
::答案是2 7 > 5 3。

Review
::回顾

Evaluate each expression.
::评估每个表达方式。

1. $\begin{aligned} 2^{2} \end{aligned}$

2. $\begin{aligned} 3^{2} \end{aligned}$

3. $\begin{aligned} 6^{2} \end{aligned}$

4. $\begin{aligned} 7^{3} \end{aligned}$

5. $\begin{aligned} 8^{4} \end{aligned}$

6. $\begin{aligned} 2^{6} \end{aligned}$

7. $\begin{aligned} 3^{5} \end{aligned}$

8. $\begin{aligned} 6^{4} \end{aligned}$

9. $\begin{aligned} 5^{3} \end{aligned}$

10. $\begin{aligned} 1^{100} \end{aligned}$

Compare each power using $\begin{aligned} < \end{aligned}$ , $\begin{aligned} > \end{aligned}$ , or $\begin{aligned} = \end{aligned}$
::使用 < 、 > 或= 来比较每个功率

11. $\begin{aligned} 4^{2} \underline{} 2^{4} \end{aligned}$

12. $\begin{aligned} 3^{2} \underline{} 1^{5} \end{aligned}$

13. $\begin{aligned} 6^{3} \underline{} 3^{6} \end{aligned}$

14. $\begin{aligned} 7^{2} \underline{} 5^{2} \end{aligned}$

15. $\begin{aligned} 8^{3} \underline{} 9^{2} \end{aligned}$

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。

Resources
::资源

Section outline

Powers ::强 权 强 权

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)

Resources ::资源

Powers
::强权强权

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)

Resources
::资源